All Questions +0 237641 Questions

Some context:
Currently, I am in 9th grade and taking Calculus BC.
I can understand competition math problems and what they are asking for but I don't really have a sense for solving them.
I should have started preparing earlier and read more ..

CInsbstnkng Oct 7, 2025

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AMC 10 I want to make AIME - PLEASE HELP

I’ve been working on this series and I’m not sure if I’m approaching it the right way:
At first glance it looks like it might behave similar to the alternating harmonic series, but the sine term is throwing me off. I’m wondering whether this read more ..

OOOOOOOO Oct 6, 2025

0

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+4

Stuck on a Tricky Series Problem

Let M⊂R∞×∞\mathcal{M} \subset \mathbb{R}^{\infty \times \infty}M⊂R∞×∞ be the set of all improperly convergent matrices whose entries are defined by the double integral:
Mij=∫0∞∫0∞sin⁡(ix)cos⁡(jy)x2+y2+ij dx dyM_{ij} = \int_0^{\infty} \int_0^{\infty} read more ..

skylerclooney Oct 1, 2025

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plz help im cooked

Alex chooses a number at random from the set {1, 2, 3, 4, 5}. Winnie also chooses a number at random from the same set. (They can choose the same number.) What is the probability that the product of their numbers is at least 4? read more ..

fof.urmum Sep 28, 2025

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Math

Bosco ●●

AnswerscorrectIy Aug 25, 2025

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Math

Suppose $f(x)$ is a function defined for all real $x$, and suppose $f$ is invertible (that is, $f^{-1}(x)$ exists for all $x$ in the range of $f$). If the graphs of $y=f(x)$ and $y=f(x^3)$ are drawn, at how many points do they intersect?

In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle CAD = 45^\circ,$ and $AD = 24,$ then find the area of triangle $ABC.$

AnswerscorrectIy Aug 25, 2025

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+943

Math

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $BC = 15,$ and $CX = 5$. What is $BX$?

booboo44 Aug 23, 2025

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+943

Math

Bosco ●

booboo44 Aug 23, 2025

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+1285

Math

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $AC = 12$, find $BC$.

In rectangle , points and lie on so that and is the midpoint of . Also, intersects at and at . The area of rectangle is 70. Find the area of triangle .
read more ..

AnswerscorrectIy Aug 23, 2025

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+67

Geometry Please answer with steps

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $AC = 12$, find $BZ$.

bqimath Aug 23, 2025

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+1285

Math

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

AnswerscorrectIy Aug 23, 2025

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+1285

Math

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

AnswerscorrectIy Aug 22, 2025

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+1285

Math

Regular hexagon ABCDEF is inscribed in rectangle PQRS. If [AFP] = 20 and [ABC] = 25, then find [ABCDEF].

AnswerscorrectIy Aug 22, 2025

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7

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+970

Math

Point P lies in regular hexagon ABCDEF such that [ABP ] = 3, [CDP] = 3, and [EFP] = 3. Compute [BCP].

gnistory Aug 21, 2025

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9

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+970

Math

gnistory Aug 21, 2025

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+490

Math

Suppose the domain of f is (-1,3). Define the function g by
g(x) = f((x + 1)(x - 2)).
What is the domain of g?

MeIdHunter Aug 20, 2025

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8

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+490

Math

There exists a polynomial $f(x)$ and a constant $k$ such that
(x^2 - 2x - 5) f(x) = 2x^4 + 19x^3 + kx^2 - 15x - 1.
What is $k?$

MeIdHunter Aug 20, 2025

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+85

Alcumus

The juniors at East HS drink a total of cartons of milk per -day week. If the average senior drinks milk at the same rate as the average junior, what is the average number of total cartons of milk the seniors drink read more ..

Trapezoid ABCD has bases \overline{AB} and \overline{CD}. The extensions of the two legs of the trapezoid intersect at $P$. If $[PBC] = 15$ and $CD = 3 \cdot AD,$ what is $[ABCD]$?

nerdboy Aug 19, 2025

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+289

Help me help

CInsbstnkng Aug 19, 2025

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+289

Need help

Trapezoid ABCD has bases \overline{AB} and \overline{CD}. The extensions of the two legs of the trapezoid intersect at $P$. If $[ABD]=8$ and $[PBC]=8$, then what is $[PAB]$?

CInsbstnkng Aug 19, 2025

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+67

Algebra

Let p(x) be defined on such that
where y is the greatest prime factor of . Express the range of p in interval notation.
Thanks and also please show the steps too!

The function f(x) is defined for 1 \le x \le 5 as follows:
f(x) = 2x + 8 if 1 \le x \le 2
f(x) = 13 - 5x if 2 < x \le 3
f(x) = 20 - 14x if 3 < x \le 4 read more ..

bqimath Aug 19, 2025

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25

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+490

Math

Given 5 colors to choose from, how many ways can we color the four unit squares of a 2\times 2 board, given that two colorings are considered the same if one is a rotation or reflection of the other? (Note that we can use the same color for more than one read more ..

MeIdHunter Aug 19, 2025

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how to solve

An equilateral triangle, a regular heptagon, a square, a 20-gon, a regular pentagon, a regular hexagon, and a regular n-gon, all with the same side length also completely surround a point. Find n.

AUnVerifedTaxPayer Aug 19, 2025

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26

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+289

help me help me

Let ABCDE be an equilateral pentagon. If the pentagon is concave, and $\angle A = \angle B = 90^{\circ},$ then what is the degree measure of $\angle E$?

CInsbstnkng Aug 18, 2025

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25

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help me

Find all ordered pairs x, y of real numbers such that x+y=10 and x^3+y^3=300.
For example, to enter the solutions (2, 4) and (-3, 9), you would enter "(2,4),(-3,9)" (without the quotation marks).

CInsbstnkng Aug 18, 2025

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25

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+289

Help help

For an integer n, the inequality
x^2 + nx + 15 < -89 - 3x^2
has no real solutions in $x$. Find the number of different possible values of $n$.

CInsbstnkng Aug 17, 2025

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25

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+289

Help me help

CInsbstnkng Aug 17, 2025

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+842

CPhill answer me

Let $m$ and $n$ be positive integers. If $m$ has exactly $7$ positive divisors, $n$ has exactly $15$ positive divisors, and $mn$ has exactly $14$ positive divisors, then how many divisors does $m^2 n^2$ have?

cooIcooIcooI17 Aug 16, 2025

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28

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Help me CPhill Now

Let $a$, $b$, $c$, and $n$ be positive integers. If $a + b + c = 19 \cdot 97$ and
a + 3n = b - 2n = \frac{c}{5n},
compute the value of $a$.

Let $A$, $B$, and $C$ be constants so that
\frac{1}{x^3-2x^2-5x+9}=\frac{A}{x+5}+\frac{B}{x-3} + \frac{C}{(x - 3)^2}
holds for all real numbers $x$ other than $-5$ and $3.$ What is $A$?

HumanBemg Aug 16, 2025

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25

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+483

Help me

For integers $a$ and $T$, $T \neq 0,$ a parabola whose general equation is $y = ax^2 + bx + c$ passes through the points $A = (0,0),$, $B = (2T,0)$, $C = (2T + 1,28).$ and $N$ Let be the sum read more ..

bIueb3rry Aug 16, 2025

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+23

NEED HELP CPhill ASAP

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

Dragononweb2.0calc Aug 16, 2025

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+417

Geometry

I have to paint one side of a wall. The wall is $15$ meters tall and $60$ meters long. Each gallon of paint covers $400$ square feet. If a foot is approximately $0.3048$ meters, then what is the smallest whole number of gallons I can buy and read more ..

nathanl6656 Aug 15, 2025

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+970

Math

Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

gnistory Aug 15, 2025

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+943

I need answer

In the diagram below, is a square. Find x[asy]
unitsize(1.5 cm);
pair A, B, C, D, P;
A = (0,3);
B = (0,0);<#> C = (3,0);<#> D = (3,3);<#> P = (0.8,1.8);<#> draw(A--B--C--D--cycle);<#> draw(A--P);<#> draw(B--P);<#> read more ..

booboo44 Aug 15, 2025

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25

6

+5

help me plz

Shown below is rectangle EFGH. Its diagonals meet at Y. Let $X$ be the foot if an altitude is dropped from $E$ to $\overline{FH}$. If EX = 24 and $GY = 30$, find the perimeter of rectangle $EFGH$.

askdjlkadjwl123 Aug 14, 2025

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27

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+289

help me now

CInsbstnkng Aug 13, 2025

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26

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+289

Help help help

Find the sum of the squares of the roots of $2x^2+4x-1=x^2-8x+3$.

CInsbstnkng Aug 11, 2025

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28

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+289

Help me

Let $m$ be a real number. If the quadratic equation $x^2+mx+4 = 2x^2 + 17x + 8$ has two distinct real roots, then what are the possible values of $m$? Express your answer in interval notation.

CInsbstnkng Aug 11, 2025

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28

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+289

Help me!

Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)

CInsbstnkng Aug 11, 2025

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28

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+289

Help

Find one ordered pair $(x,y)$ of real numbers such that $x + y = 10$ and $x^3 + y^3 = 162 + x^2 + y^2.$

CInsbstnkng Aug 11, 2025

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26

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+289

help help

In triangle ABC, the angle bisector of angle BAC meets BC at D, such that AD = AB. Line segment AD is extended to E, such that angle DBE = angle BAD = 17 degrees. Find angle ABD.

CInsbstnkng Aug 10, 2025

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+289

HELP!!!

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

CInsbstnkng Aug 9, 2025

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26

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+289

help me now

We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude to $\triangle ABC$. If $AC = 8,$ $BK = 2$, and $CK = 3,$ then what is $AB$?

CInsbstnkng Aug 8, 2025

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25

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+289

help!!!

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$ read more ..

CInsbstnkng Aug 8, 2025

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+289

help me plz

Lines $XQ$ and $XR$ are tangent to a circle, as shown below. If $\angle XTU = 47^\circ$ and $\angle QAU = 65^\circ,$ then find $\angle QXR,$ in degrees.

CInsbstnkng Aug 8, 2025

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+289

help me!

For an integer $n,$ let $f(n)$ be the remainder when $n^8 + n^{16}$ is divided by $5.$ Compute $f(0) + f(1) + f(2) + f(3) + f(4).$

CInsbstnkng Aug 7, 2025

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+289

help help help

For a certain positive integer $n$, the number $n^{6873}$ leaves a remainder of $3$ when divided by $23.$ What remainder does $n$ leave when divided by $23$?

CInsbstnkng Aug 7, 2025

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26

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+289

I need help

Let a and b be positive real numbers such that a + b = 1. Find the set of all possible values of a/b.

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

CInsbstnkng Aug 5, 2025

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+289

Help me

CInsbstnkng Aug 5, 2025

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Help help

The tangent to the circumcircle of triangle $WXY$ at $X$ is drawn, and the line through $W$ that is parallel to this tangent intersects $\overline{XY}$ at $Z.$ If $XZ = 5$ and $YZ = 2,$ find $WX.$

CInsbstnkng Aug 3, 2025

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Need help

In triangle $ABC,$ $\angle C = 90^\circ.$ A semicircle is constructed along side $\overline{AC}$ that is tangent to $\overline{BC}$ and $\overline{AB}.$ If the radius of the semicircle is equal to $1$ and $AB = 2$, then find $AC.$

CInsbstnkng Aug 3, 2025

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Help me help

Let $k$ be a positive real number. The line $x + y = 3 + k$ and the circle $x^2 + y^2 = k$ are drawn. Find $k$ so that the line is tangent to the circle.

CInsbstnkng Aug 3, 2025

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+67

Algebra

A sector of a circle is shown below. The sector has an area of $60 \pi.$ What is the radius of the circle?

bqimath Aug 2, 2025

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Plz help

Trapezoid $HGFE$ is inscribed in a circle, with $\overline{EF} \parallel \overline{GH}$. If arc $GH$ is $66$ degrees, arc $EH$ is $x^2 + 8x$ degrees, and arc $FG$ is $25 + 16x$ degrees, where $x > 0,$ find arc $EPF$, in degrees.

CInsbstnkng Aug 2, 2025

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Help

There is a group of five children, where two of the children are twins. In how many ways can I distribute $12$ identical pieces of candy to the children, if the twins must get an equal amount of candy?

CInsbstnkng Aug 2, 2025

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Need help (it's really me!!!)

How many solutions are there to the equation
u + v + w + x + y + z = 6
where $u,$ $v,$ $w,$ $x,$ $y,$ and $z$ are nonnegative integers, and $x$ is at most $8$ and $y$ is even?

CInsbstnkng Aug 2, 2025

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Help!!! (really me)

In the complex plane, z, z2, z3, form, in some order, three of the vertices of a non-degenerate square. Enter all possible areas of the square, separated by commas.

CInsbstnkng Aug 2, 2025

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5

+62

HELP!!! (From the real Clnsbstnkng)

The function f(x) is defined for 1 \le x \le 5 as follows:
f(x) = 2x + 8 if 1 \le x \le 2
f(x) = 13 - 5x if 2 < x \le 3
f(x) = 20 - 14x if 3 < x \le 4 read more ..

Clnsbstnkng Aug 2, 2025

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+289

HELP!!!

CInsbstnkng Aug 1, 2025

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help

Find constants A and B such that
(2x + 11)/(x^2 - 6x + 5) = A/(x - 3) + B/(x + 4)
for all x such that x neq 7 and x neq 8

CInsbstnkng Aug 1, 2025

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Help me

If z is a complex number such that z + 1/z = 3, what is the value of z^2 + 1/z^2?

Let z1 and z2 be two complex numbers such that |z1| = 5 and (z1/z2)+(z2/z1)=0.
Find z1+z2.

CInsbstnkng Aug 1, 2025

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HELP HELP

George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for , and wants each husband to be seated opposite his wife. How many seating arrangements can he make, read more ..

CInsbstnkng Aug 1, 2025

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+67

Counting & Probability

An ellipse has its foci at (-1, -1) and (-1, -3).
Given that it passes through the point (0, 2), its equation can be written in the form ((x-h)2/a2)/((y-k)2/b2)=1 where a, b, h, k are constants, and a and b are positive. Find a+k.

bqimath Aug 1, 2025

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+289

HELP ME

CInsbstnkng Jul 31, 2025

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HELP

Points $A$ and $B$ are on side $\overline{YZ}$ of rectangle $WXYZ$ such that $\overline{WA}$ and $\overline{WB}$ trisect $\angle ZWX$. If $BY = 2$ and $AZ = 4$, then what is the area of rectangle $WXYZ$?

In rectangle $WXYZ$, $A$ is on side $\overline{WX}$, $B$ is on side $\overline{YZ}$, and $C$ is on side $\overline{XY}$. If $AX = 2$, $BY = 16$, $\angle ACB = 90^\circ$, and $CY = 2 \cdot CX$, then find $AB$.

CInsbstnkng Jul 31, 2025

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+289

help help

CInsbstnkng Jul 31, 2025

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help me!

A sphere of radius 4 inches is inscribed in a cone with a base of radius 4 inches. In inches, what is the height of the cone? Express your answer as a decimal to the nearest tenth

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

CInsbstnkng Jul 31, 2025

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HELP!!!

Suppose a does not equal 0. Compute log base 8a (4^b) in terms of a and b.

CInsbstnkng Jul 31, 2025

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Help!

The ellipse shown below is defined by the equation PF1+PF2=d.
Find d.

CInsbstnkng Jul 31, 2025

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+62

HELP!!!

Rectangle ADEH is divided into rectangles ABGH, BCFG, and CDEF. If $AC = 15,$ $GE = 15,$ $[ABGH] = 8,$ and $[CDEF] = 8,$ find $[BCFG].$

Clnsbstnkng Jul 30, 2025

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help help

CInsbstnkng Jul 30, 2025

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help me

In the diagram, XY = 2\cdot AX = 2\cdot YB and $\dfrac{AZ}{BX} = \dfrac{2}{5}$. Find $\dfrac{[YBC]}{[ZYC]}$.

CInsbstnkng Jul 30, 2025

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plz help

Find the number of solutions to
x_1 + x_2 + x_3 + x_4 + x_5 \le 1
in nonnegative integers.

In how many ways can you distribute $8$ indistinguishable balls among $6$ distinguishable boxes, if at least three of the boxes must be empty?

CInsbstnkng Jul 29, 2025

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Help help!

Point X is on side \overline{AC} of triangle $ABC$ such that $\angle AXB =\angle ABX + 20^\circ$, and $\angle ABC - \angle BAC = 36^\circ$. Find $\angle XBC$ in degrees.

CInsbstnkng Jul 29, 2025

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Help me

Let $a_1,$ $a_2,$ $a_3,$ $\dots,$ $a_8,$ $a_9,$ $a_{10}$ be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 = 5$ and $a_2 + a_4 + a_6 + a_8 + a_{10} = -2$, then find $a_1$.

CInsbstnkng Jul 29, 2025

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Help!! real

A sphere of radius inches is inscribed in a cone with a base of radius inches. In inches, what is the height of the cone? Express your answer as a decimal to the nearest tenth. read more ..

CInsbstnkng Jul 29, 2025

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Geometry from alcumus

●●●●●●●

bqimath Jul 29, 2025

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Help me plz

Triangle DEF is the medial triangle of triangle ABC, and triangle $XYZ$ is the medial triangle of triangle $DEF.$ If the area of triangle $XYZ$ is $8$, what is the area of triangle $ABC?$

In triangle ABC, \angle A = 90^\circ. Altitude $\overline{AP},$ angle bisector $\overline{AQ},$ and median $\overline{AR}$ are drawn. If $PQ = 3$ and $QC = 4,$ find $AR.$

CInsbstnkng Jul 29, 2025

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Need help now

An ellipse has its foci at (-1, -1) and (-1, -3).
Given that it passes through the point (4, -2), its equation can be written in the form ((x-h)2/a2)/((y-k)2/b2)=1 where a, b, h, k are constants, and a and b are positive. Find a+k.

CInsbstnkng Jul 29, 2025

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7

+62

HELP!!! (From the real Clnsbstnkng)

●●●●●●●

Clnsbstnkng Jul 29, 2025

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Help!

Find the coefficient of u^2*v^9 in (2u - 3*v^3)*5*(3u^3 - 2v^5)^3.

CInsbstnkng Jul 28, 2025

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Help help

Find the constant term in the expansion of (2z - 1/sqrt(z))*5*(z - 1/sqrt(z))^3.

CInsbstnkng Jul 28, 2025

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Help

Four cubes of volumes $1 \text{ cm}^3$, $2 \text{ cm}^3$, $3 \text{ cm}^3$, and $4 \text{ cm}^3$ are glued together at their faces. What is the number of square centimeters in the smallest possible surface area of the resulting solid figure?

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

CInsbstnkng Jul 28, 2025

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Help!!

Triangle ABC is congruent to triangle DEF, but triangle ABC is not congruent to triangle XYZ. Which of the following relationships cannot hold?
ABC and DEF have the same area
ABC and DEF have the same perimeter
ABC read more ..

CInsbstnkng Jul 28, 2025

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Help with geo

In pentagon ABCDE, \overline{AC} bisects \angle BCE and $\overline{AD}$ bisects $\angle BDE.$ If $\angle CBD = 30^\circ$ and $\angle CED = 60^\circ,$ then find $\angle CAD,$ in degrees.

CInsbstnkng Jul 28, 2025

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Need help

Compute the smallest positive integer n such that n+i, (n+i)*2, and (n+i)*3 are the vertices of a triangle in the complex plane whose area is greater than 2015.

CInsbstnkng Jul 28, 2025

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Plz help!!!

In the complex plane, z, z2, z3, form, in some order, three of the vertices of a non-degenerate square. Enter all possible areas of the square, separated by commas.

CInsbstnkng Jul 28, 2025

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+62

HELP!!! (From the OG Clnsbstnkng)

In the complex plane, z, z2, z3, form, in some order, three of the vertices of a non-degenerate square. Enter all possible areas of the square, separated by commas.

Clnsbstnkng Jul 28, 2025

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+62

HELP!!!

Circle O has radius 5. Segment $\overline{AS}$ has length $5$ and is tangent to circle $O$ at $A$. If $P$ is the projection of A onto OS, find length PS.

Clnsbstnkng Jul 28, 2025

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Plz help

Two points, A and B, are on the pavement 6 inches apart. Cassidy the caterpillar traces out a path walking along all the points that are equidistant to as from . Cassidy has red paint on her feet, so she literally traces out the whole path! read more ..

CInsbstnkng Jul 27, 2025

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Help me

Let f(x) be a polynomial. Find the remainder when f(x) is divided by x(x - 1)(x - 2), if f(0) = 0, f(1) = 1, and f(2) = 4.

CInsbstnkng Jul 27, 2025

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Plz help

In triangle ABC, the angle bisector of angle BAC meets BC at D, such that AD = AB. Line segment AD is extended to E, such that angle DBE = angle BAD = 17 degrees. Find angle ABD.

CInsbstnkng Jul 27, 2025

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Help help help

Recall that a partition of a positive integer n means a way of writing n as the sum of some positive integers, where the order of the parts does not matter. For example, there are five partitions of 4:
4
3 + 1
2 + 2 read more ..

CInsbstnkng Jul 27, 2025

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Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of triangle ABC?
A is at (0,0), B is at (1,0), and C is at (1,1).

CInsbstnkng Jul 27, 2025

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Need help

In the complex plane, z, z2, z3, form, in some order, three of the vertices of a non-degenerate square. Enter all possible areas of the square, separated by commas.

CInsbstnkng Jul 27, 2025

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HELP!!!

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BD=12, AE=8, and BF=3+AE. Find AB.

Clnsbstnkng Jul 26, 2025

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Help CPHILL!!!

Compute the smallest positive integer n such that n+i, (n+i)2, and (n+i)3 are the vertices of a triangle in the complex plane whose area is greater than 2015.

CInsbstnkng Jul 26, 2025

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6

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HELP!!!

In the diagram, \overline{AD} is an altitude, \overline{BE} is a median, and \overline{AD}, \overline{BE}, and \overline{CF} are concurrent. If AE = 11, BE = 9, and AD = 6\sqrt{2}, what is AF?

Clnsbstnkng Jul 25, 2025

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Help!

How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

CInsbstnkng Jul 25, 2025

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1

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Help

How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?

CInsbstnkng Jul 25, 2025

0

4

1

+289

Help

Karlanna places 600 marbles into m total boxes such that each box contains an equal number of marbles. There is more than one box, and each box contains more than one marble. For how many values of m can this be done?

CInsbstnkng Jul 25, 2025

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Number Theory

Right triangles ABC, ADE, EFG, and GHC are all similar. Also, $[ADE] = 1,$ $[EFG] = 1,$ and $[GHC] = 1.$ Find the area of the red region.

bqimath Jul 25, 2025

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Help help

CInsbstnkng Jul 25, 2025

-1

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Help me

Let f(x) be a polynomial such that f(0) = 4, f(1) = 8, and f(2) = -6. Find the remainder when f(x) is divided by x.

CInsbstnkng Jul 24, 2025

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3

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Help!

The polynomial
g(x) = x^3 - x^2 - (m^2 + m + 18) x + 2m^2 - 14m - 6
is divisible by x - 5 and all of its zeroes are integers. Find all possible values of m.

CInsbstnkng Jul 24, 2025

0

2

4

+62

HELP!!!

Let z be a complex number such that |z|=1.
Find the maximum value of |1+z|+|1-z+z2|.

Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.

Clnsbstnkng Jul 24, 2025

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6

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+289

Help help

CInsbstnkng Jul 24, 2025

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Help me plz

A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5).
Find the number of permutations (x_1, x_2, \dots, x_8) read more ..

Find the number of sequences (a_1, a_2, a_3, \dots, a_8) such that:
* a_i \in \{1, 2, 3, 4, 5, 6, 7, 8\} for all 1 \le i \le 8.
* Every number1, 2, 3, 4, 5, 6, 7, 8 appears at least once in the sequence.

CInsbstnkng Jul 24, 2025

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+289

Help help

A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit.
You are dealt a hand of read more ..

CInsbstnkng Jul 24, 2025

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7

0

+289

Plz help

CInsbstnkng Jul 24, 2025

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Help

If an ant crawls from one corner to the other corner of a 3 \times 5 rectangle, then it will cross through seven squares.
If the ant crawls from one corner to the other corner of a 4 \times 4 rectangle, then how many squares will it read more ..

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

CInsbstnkng Jul 24, 2025

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Help

A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The circle is x^2 + y^2 = 1, and the line is y = x.

CInsbstnkng Jul 24, 2025

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Help plz

Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $11,$ and the slope of line segment $\overline{OQ}$ is $13.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

CInsbstnkng Jul 24, 2025

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+289

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CInsbstnkng Jul 24, 2025

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Help meee

How many 7-digit sequences have a digit that appears at least 6 times?
(For example, 3339333 and 0200000 are two such sequences. A sequence is allowed to begin with 0.)

CInsbstnkng Jul 24, 2025

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4

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+289

Need help

Triangle ABC has side lengths AB = 8, BC = 10, and AC = 12. Find \dfrac{\cos A}{\cos B}.

Find the number of paths from A to B in the grid below, so that
* Each step is down or to the right.
* The path cannot pass through any point more than once.

An example path is shown.
The grid is 3 by 3, with read more ..

CInsbstnkng Jul 24, 2025

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4

1

+289

Help me

Let z be a complex number such that |z|=1.
Find the maximum value of |1+z|+|1-z+z2|.

CInsbstnkng Jul 24, 2025

0

4

3

+62

HELP!!!

Find the number of ordered pairs $(a,b)$ of integers such that
\frac{a + 2}{a + 1} = \frac{b}{12}.

Clnsbstnkng Jul 23, 2025

+1

3

4

+842

Help math

The roots of Ax^2+Bx+1 are the same as the roots of x^2- 3x + 5 + x^2 - 2x + 1. What is A+B?

cooIcooIcooI17 Jul 22, 2025

0

2

+1733

Help algebra

Let $G$ be the center of equilateral triangle $XYZ.$ A dilation centered at $G$ with scale factor $-1$ is applied to triangle $XYZ,$ to obtain triangle $X'Y'Z'.$ Let $A$ be the area of the region that is contained in both triangles $XYZ$ and $X'Y'Z'.$ Find read more ..

parmen Jul 22, 2025

0

4

0

+1004

help me

Hi6942O Jul 22, 2025

0

2

1

+1524

Need help

Find the smallest positive integer $N$ such that
N &\equiv 2 \pmod{3}, \\
N &\equiv 2 \pmod{7}, \\
N &\equiv 2 \pmod{10}.

Find the number of ways of filling in the squares of a 3 \times 3 grid so that:
* Each square contains a 0 or a 1.
* The sum of the numbers in each row and each column is at most 1.
An example is shown below.

learnmgcat Jul 22, 2025

0

7

0

+190

help math

HumanBemg Jul 22, 2025

0

3

2

+62

HELP!!!

If z is a complex number such that z+z-1=√3,
what is the value of z2010+z-2010?

Let \omega be a nonreal root of x^3 = 1. Compute
(1 - \omega + \omega^2)^6 + (1 + \omega - \omega^2)^6.

Clnsbstnkng Jul 22, 2025

0

3

1

+190

help me

A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3)$ is a permutation of (1,2,3,4,5).
Let \pi = (x_1,x_2,x_3,\dots,x_n) be a permutation read more ..

HumanBemg Jul 21, 2025

0

1

4

0

+1285

Math

AnswerscorrectIy Jul 21, 2025

0

1

3

1

+1285

Math

Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)

Let z1 and z2 be two complex numbers such that |z1| = 5 and (z1/z2)+(z2/z1)=1.
Find |z1-z2|2.
Note: The answer is not 75.

AnswerscorrectIy Jul 21, 2025

0

2

3

+62

HELP!!!

In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may sit next to each other in the same row, but no child may sit directly in front of their sibling?

Clnsbstnkng Jul 21, 2025

0

2

+483

Math

Let x, y, and z be real numbers. If x^2 + y^2 + z^2 = 1, then find the maximum value of
3x + 4y + 5z + x^3 + \frac{4x^2 y}{z} + \frac{z^5}{xy^2}

bIueb3rry Jul 20, 2025

+1

5

2

1

+483

Algebra

Let x_1, x_2, \dots, x_{100} be real numbers. If
x_1 + 2x_2 + \dots + 100x_{100} = 1,
then find the minimum value of x_1/1 + x_2/2 + \dots + x_{100}/100.

bIueb3rry Jul 12, 2025

0

2

1

+483

Algebra

In rectangle $WXYZ$, $A$ is on side $\overline{WX}$, $B$ is on side $\overline{YZ}$, and $C$ is on side $\overline{XY}$. If $AX = 15$, $BY = 20$, $\angle ACB= 60^\circ$, and $CY = 3 \cdot CX$, then find $AB$.

bIueb3rry Jul 11, 2025

+2

5

2

+1285

Math

In my derivation for the tangential normal expression for an acceleration vector, I got that acceleration equals:
However, I also learned it can be expressed as:
How are these two expressions read more ..

AnswerscorrectIy Jul 7, 2025

0

8

1

+12

help with simple multi-variable calc/physics concept

In the diagram below, ABCD is a square, DE = EF = FG = 2, GH = 1, HB = 5, and \angle DEF = \angle EFG = \angle FGH = \angle GHB = 60^\circ. Find the area of the shaded region.

albertthesav Jul 6, 2025

0

1

2

1

+1285

Math

In triangle ABO, M is the midpoint of \overline{AB}. Points C, D, and N lie on line segments $\overline{AO},$ $\overline{BO},$ and $\overline{MO}$ extended, respectively, so that $DO = BO,$ $NO = MO,$ and $CO = AO.$ If $K$ is the area of triangle read more ..

AnswerscorrectIy Jul 3, 2025

0

6

2

1

+1285

Math

Franklin the Fly lives at one corner of a cube of side length 1. He doesn't like to wander too far from home; in fact, Franklin is only willing to travel to a point on the cube if he can reach it by crawling a distance of at most \sqrt{5} along the read more ..

AnswerscorrectIy Jul 3, 2025

+1

4

2

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Math

Five positive integers are written around a circle so that no two or three adjacent number have a sum
divisible by 3. In this collection the number of numbers divisible by 3 is .
(A) 0 (B) 1 (C) 2 (D) 3

AnswerscorrectIy Jul 3, 2025

+1

2

+10

Pls help with this question

The equation
holds for exactly two real values of and their sum is . Find

Brilliance Jul 3, 2025

+1

3

2

+215

Hard Math Question!

Find all values of the real number $a$ so that the four complex roots of \[z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0\]
form the vertices of a parallelogram in the complex plane. Enter all the values, separated by commas.
I read more ..

xud33 Jul 2, 2025

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26

1

+5

In how many ways can I choose 2 cooks on a backpacking trip with 8 people if any of the 8 people may be cooks?

mercurym999 ●

ansonh138 Jun 24, 2025

+1

29

1

+24

complex number roots

What is the distance between the points (\cos 37^{\circ}, \sin 37^{\circ}) and (\cos 127^{\circ}, \sin 127^{\circ})?

mercurym999 Jun 21, 2025

+1

3

25

2

+943

Geometry

Find an acute angle A such that \sin 4A = \sin A + \sin 2A. Express your answer in degrees.

booboo44 Jun 20, 2025

0

2

25

5

+943

Geometry

In the complex plane, let $S$ be the set of complex numbers $z$ such that \[\left| z + \frac{1}{z} \right| \le 2.\]
Find the area of $S.$
Thanks

booboo44 Jun 20, 2025

+1

27

3

+24

complex number geometry help

Find the number of arithmetic sequences such that:
* The arithmetic sequence contains three terms
* All the terms are integers in \{0, 1, 2, \dots, 10\}
* The sum of the terms is 9.

mercurym999 Jun 19, 2025

0

4

25

3

+1285

Counting

Bosco ●●●

AnswerscorrectIy Jun 19, 2025

0

26

2

+15

Geometery

A cube has side length . Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.) read more ..

Let x_1, x_2, \dots, x_{100} be real numbers. If
x_1 + 2x_2 + \dots + 100x_{100} = 1,
then find the minimum value of x_1/1 + x_2/2 + \dots + x_{100}/100.

Ibrahim097YT Jun 19, 2025

0

25

4

+120

Algebra

Given the function y = f(x) = x2.
a) From the graph of the function y = f(x) (Figure 4), indicate the increasing and decreasing intervals of the given function.
b) Calculate the derivative of f'(x) and examine the sign of f'(x).
c) read more ..

coIinqiu Jun 12, 2025

0

2

26

1

+4

Given the function y = f(x) = x2.

What is the area, in square units, of a regular hexagon inscribed in a circle whose area is square units? Express your answer in simplest radical form.

molisa Jun 12, 2025

+1

25

5

+3

Help plz enter explaination

Arrange the following numbers in increasing order (smallest first, biggest last):
1.
2.
3. read more ..

ilrelyt Jun 12, 2025

+1

32

2

+17

Algebra

Consider the set
S = {1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 18, 23, 24, ..., 12345678},
which consists of all positive integers whose digits strictly increase from left to right, and the digits are from 1 to 8. This set is read more ..

Chiccen Jun 11, 2025

0

3

26

1

+1285

Counting

Find the number of ways filling in a 4 \times 4 grid, such that
* Each cell contains a 0 or a 1.
* The sum of the numbers in each row and each column is at least 2.

An example is shown below.
0110 read more ..

AnswerscorrectIy Jun 11, 2025

0

2

26

1

+1285

Counting

AnswerscorrectIy Jun 11, 2025

0

25

6

+5

How many 6-digit sequences have a digit that appears at least 5 times?

In pentagon ABCDE, \overline{AC} bisects \angle BCE and $\overline{AD}$ bisects $\angle BDE.$ If $\angle CBD = 30^\circ$ and $\angle CED = 60^\circ,$ then find $\angle CAD,$ in degrees.

Beecomen1983 Jun 10, 2025

0

1

26

2

+842

Geometry

Not sure if anyone remembers me anymore; I randomly remembered about this app and the good times I had on it asking for help and helping others with their math, so just wanted to say hi :D hope everyone is doing well! and if you are reaeding this, have read more ..

cooIcooIcooI17 May 29, 2025

+2

30

3

+707

Hello Guys!

Circle O has radius 5. Segment $\overline{AS}$ has length $5$ and is tangent to circle $O$ at $A$. If $P$ is the projection of A onto OS, find length PS.

Nirvana May 28, 2025

+1

26

1

+190

Geometry

Two runners start at vertices A and B simultaneously and run clockwise around the perimeter of square ABCD at the same speed. A drone flies so that it is always at the midpoint of the two runners. What is the minimum distance between the two runners? read more ..

HumanBemg May 28, 2025

0

3

25

1

+483

Geometry

Let x and y be real numbers such that x^2 + y^2 = 2x + 4y. Find the largest possible value of x + y.

bIueb3rry May 27, 2025

0

2

25

2

+343

Geometry

Two black lines pass through the origin, as shown below. If their slopes are 2 and 8, then find the slope of the red line that bisects the acute angle between these lines.

jaekg May 27, 2025

0

2

25

2

+810

Geometry

What is the sine of an acute angle whose cosine is \frac{5}{7}?

crimefightingvigiI May 27, 2025

0

2

25

2

+1285

Geometry

NotThatSmart ●●

AnswerscorrectIy May 27, 2025

0

32

1

+8

In the diagram below, we have angle ABC= angle CAB= angle DEB= angle BD. Find CA.

If x and y are integers such that 12^2 \cdot 18^3 = xy, find x+y.

greenflag May 26, 2025

-1

26

2

+1285

Algebra

In the figure, point is the center of the circle, the measure of angle RTB is 28 degrees, and the measure of angle ROB is three times the measure of angle SOT. What is the measure of minor arc RS , in degrees?
I read more ..

AnswerscorrectIy May 25, 2025

+2

4

27

1

+23

In the figure, point is the center of the circle, the measure of angle RTB is 28 degrees, and the measure of angle ROB is three times the

CPhill ●

Dragononweb2.0calc May 24, 2025

-1

26

2

+9

Probability

You have a standard deck of 52 cards.
(a) What is the probability that when you draw 5 cards at random, you draw a straight (any 5 cards in order)? Order doesn't matter (e.g. A-2-3-4-5 is the same as 3-5-A-4-2.)
(b) read more ..

How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4?

serenitypie May 23, 2025

0

3

2

+190

Counting

A teacher was using a random group generator to split 24 different students, including A, B, and C, in 8 groups of 3. What is the probability that the 3 students, A, B, and C, are in the same group?

HumanBemg May 23, 2025

0

5

2

+9

Probability

We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen). What is the probability that we chose a multiple of 5?

serenitypie May 22, 2025

+1

4

2

+190

Counting

Which of these is the general solution for ? Is it or is it ? I'm so confused, is it because it is less?

HumanBemg May 22, 2025

+1

3

2

+5

Which of these is the general solution for -sqrt3sectheta = 2?

Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.

panickedferret May 19, 2025

0

8

1

+483

Counting

In parallelogram EFGH, let M be the point on \overline{EF} such that FM:ME = 1:1, and let N be the point on \overline{EH} such that HN:NE = 1:1. Line segments \overline{FH} and \overline{GM} intersect at P, and line segments \overline{FH} and GN intersect read more ..

bIueb3rry May 19, 2025

0

5

1

+53

Geometry

Ibrahin097YT May 15, 2025

0

3

2

+4

What is the focus of the parabola? y=14x2−x−1

A rectangle in the coordinate plane is shown below. A line with slope 2 splits this rectangle into two regions with equal area. What is the y-intercept of this line?
The rectangle has coordinates (-1,-1), (-1,1), (1,1), read more ..

smithale May 15, 2025

-1

7

1

+53

Geometry

Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $11,$ and the slope of line segment $\overline{OQ}$ is $13.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: read more ..

Ibrahin097YT May 13, 2025

-1

10

1

+53

Geometry

A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The circle is x^2 + y^2 = 1, and the line is y = x.

Ibrahin097YT May 13, 2025

-1

5

1

+53

Geometry

Chords UV, WX, and YZ of a circle are parallel. The distance between chords UV and WX is 1, and the distance between chords WX and YZ is also 1. If UV =6 and YZ = 4, then find WX.

Ibrahin097YT May 13, 2025

0

4

1

+53

Geometry

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

Ibrahin097YT May 13, 2025

-1

11

0

+53

Geometry

Ibrahin097YT May 13, 2025

0

13

0

+53

Geometry

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

Ibrahin097YT May 13, 2025

0

6

2

+456

Counting

How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

The following cards are split into four piles at random, so that every pile contains the same number of cards. What is the probability that every pile contains an Ace?
The cards are:
Ace of clubs
Ace of diamonds read more ..

Jasmo May 10, 2025

0

1

4

1

+483

Counting

A bag contains red and blue tiles. Each tile has a number from the set \{-1, 0, 1\} written on it. I want to arrange 7 of these tiles in a row, so that the numbers on any three consecutive tiles sum to 3. In how many ways can this be done, assuming that read more ..

bIueb3rry May 10, 2025

0

3

1

+483

Counting

Find the number of ways of filling in the squares of a 3 \times 3 grid so that:
* Each square contains a 0 or a 1.
* The sum of the numbers in each row and each column is at most 1.
An example is shown below.

bIueb3rry May 10, 2025

-1

12

0

+53

i need answer now

Ibrahin097YT May 10, 2025

-1

13

1

+53

plz help now

Find the number of paths from A to B in the grid below, so that
* Each step is down or to the right.
* The path cannot pass through any point more than once.

An example path is shown.
The grid is 3 by 3, with read more ..

If an ant crawls from one corner to the other corner of a 3 \times 5 rectangle, then it will cross through seven squares.
If the ant crawls from one corner to the other corner of a 15 \times 16 rectangle, then how many squares will read more ..

Ibrahin097YT May 10, 2025

-1

8

1

+53

help me help help

A survey conducted among 150 high school students revealed that

* 68 students like Math
* 85 students like English
* 55 students like History
* 20 students like both Math and English
* 15 students like both Math and read more ..

Ibrahin097YT May 10, 2025

0

5

3

+53

help really need help

What is the value of the expression. Express your asnwer as a common fraction.

Ibrahin097YT May 10, 2025

0

10

1

+15

HELP PLEASE ASAP!!!!

Let ABC be a triangle with side lengths AB = 5, BC = 6, and AC = 9. What is the area of the triangle with side lengths \tan A, \tan B, and \tan C?

Ibrahim097YT May 10, 2025

-1

3

2

+1808

Geometry

Acute triangle ABC has \sin A = \dfrac{2}{5} and \sin B = \dfrac{2}{\sqrt{7}}. What is \sin C?

blackpanther May 9, 2025

-1

5

1

+1808

Geometry

Find the area of the triangle with side lengths 4, 5, and 8.

blackpanther May 9, 2025

0

7

2

+1808

Geometry

Triangle ABC has side lengths AB = 8, BC = 10, and AC = 12. Find \dfrac{\cos A}{\cos B}.

blackpanther May 9, 2025

0

9

0

+1808

Geometry

blackpanther May 9, 2025

0

13

0

+608

Counting

A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit.
You are dealt a hand of read more ..

onyuIee May 9, 2025

0

8

0

+608

Counting

Find the number of sequences (a_1, a_2, a_3, \dots, a_8) such that:
* a_i \in \{1, 2, 3, 4, 5, 6, 7, 8\} for all 1 \le i \le 8.
* Every number1, 2, 3, 4, 5, 6, 7, 8 appears at least once in the sequence.

onyuIee May 9, 2025

0

9

0

+608

Counting

A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5).
Find the number of permutations (x_1, x_2, \dots, x_8) read more ..

onyuIee May 9, 2025

0

12

0

+53

How many subsets of \{0, 1, \dots, 9\} have the property

How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4?

Ibrahin097YT May 9, 2025

0

8

0

+53

We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen).

We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen). What is the probability that we chose a multiple of 5?

Ibrahin097YT May 9, 2025

0

12

0

+53

Find the number of ways of arranging one A, two Bs, three Cs, and four Ds

Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.

Ibrahin097YT May 9, 2025

0

9

1

+4

Trigonometric Laws and Identities

Two students are calculating the exact value of tan(7pi/12)? Choose one of the following to methods to solve.
-Student A says 7pi/12 is half of 7pi/6, and they will use the half-angle identity and the unit of circle to solve.
-Student B says read more ..

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BC = 12, AC = 18 and BF = 3 \cdot AF. Find AB.

blueberiii May 9, 2025

0

10

0

+53

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors

Ibrahin097YT May 9, 2025

+1

9

1

+15

What is the value of the expression shown? Express your answer as a common fraction.

How many 7-digit sequences have a digit that appears at least 6 times?
(For example, 3339333 and 0200000 are two such sequences. A sequence is allowed to begin with 0.)

Ibrahim097YT May 9, 2025

+1

4

0

+1271

Counting

Akhain1 May 1, 2025

0

3

4

1

+842

Counting

How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?

How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

cooIcooIcooI17 May 1, 2025

0

1

5

0

+842

Counting

cooIcooIcooI17 May 1, 2025

0

12

0

+56

Geometry

In the diagram, \overline{AD} is an altitude, \overline{BE} is a median, and \overline{AD}, \overline{BE}, and \overline{CF} are concurrent. If AE = 11, BE = 9, and AD = 6\sqrt{2}, what is AF?

PikaPika989 Apr 26, 2025

0

11

1

+56

Geometry

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BD=12, AE=8, and BF=3+AE. Find AB.

If a drone has Max ISO 44000, 12-Bit DNG, how will its night 6K photos be?

PikaPika989 Apr 26, 2025

0

15

1

+4

Calculate the best ISO for drone night scenes

How many squares in the plane have at least two points in the lattice below as vertices?

rainyxyb Apr 24, 2025

0

6

1

+1285

Counting

At a meeting, two scientists, two mathematicians, two historians, and two artists are to be seated around a circular table. In how many ways can they be seated so that all four pairs of people from the same discipline are seated together?

AnswerscorrectIy Apr 23, 2025

0

4

2

+1285

Counting

What is the equation of the reflection of the line y = 2x + 1 across the line y = 5x - 3? Give your answer in slope-intercept form.

AnswerscorrectIy Apr 23, 2025

0

10

1

+56

Help me plz

In \triangle ABC, we have AB = AC = 4 and \angle BAC = 45^\circ. If M is the midpoint of BC, then find AM^2.

PikaPika989 Apr 23, 2025

0

5

1

+56

Help need help

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of triangle ABC?
A is at (0,0), B is at (1,0), and C is at (1,1).

PikaPika989 Apr 23, 2025

0

8

1

+56

Help geometry

In the complex plane, z, z2 , z3 form, in some order, three of the vertices of a non-degenerate square. Enter all possible areas of the square, separated by commas.

PikaPika989 Apr 23, 2025

+1

4

1

+16

Super Hard Algebra 2/Precalculus Question Help Please

Without knowing the relationship between h and x, how is this possible?

PikaPika999 Apr 23, 2025

0

3

1

+4

Any help appreciated - preferably in simple language for a non mathematician

Let S be a set of distinct integers. What is the smallest number of elements that S must contain, to ensure that S has a nonempty subset, where the sum of the elements in the subset is divisible by 2?

dadtryingtoHelp Apr 22, 2025

0

10

1

+56

Counting

Vera has 20 white socks, 21 black socks, 22 brown socks, 23 blue socks, 24 red socks, and 25 green socks. How many socks (at a minimum) must she pull out of her sock drawer to ensure at least six matching pairs of different colors?

PikaPika989 Apr 22, 2025

0

11

1

+56

Counting

Recall that a partition of a positive integer n means a way of writing n as the sum of some positive integers, where the order of the parts does not matter. For example, there are five partitions of 4:
4
3 + 1
2 + 2 read more ..

PikaPika989 Apr 21, 2025

0

14

0

+977

Counting

magenta Apr 20, 2025

0

10

1

+977

Counting

Find the number of ways filling in a 4 \times 4 grid, such that
* Each cell contains a 0 or a 1.
* The sum of the numbers in each row and each column is at least 2.

An example is shown below.
0110 read more ..

Consider the set
S = {1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 18, 23, 24, ..., 12345678},
which consists of all positive integers whose digits strictly increase from left to right, and the digits are from 1 to 8. This set is read more ..

magenta Apr 20, 2025

0

15

0

+977

Counting

magenta Apr 20, 2025

0

2

5

0

+483

Counting

Find the number of subsets of
S = \{1, 3, 8, 17, 30, 36, 47, 58\},
so that the sum of the elements in the subset is less than 20. (Note that for the empty subset, we take the sum of the elements as 0.)

bIueb3rry Apr 19, 2025

0

1

10

0

+483

Counting

Find the number of subsets of
S = \{1, 3, 8, 17, 30, 36, 47, 58\},
so that the sum of the elements in the subset is a multiple of 5. (Note that for the empty subset, we take the sum of the elements as 0.)

bIueb3rry Apr 19, 2025

0

10

0

+608

Counting

Adam the ant starts at (0,0). Each minute, he flips a fair coin. If he flips heads, he moves 1 unit up; if he flips tails, he moves 1 unit right.
Betty the beetle starts at (1,1). Each minute, she flips a fair coin. If she flips heads, read more ..

onyuIee Apr 19, 2025

0

11

1

+608

Counting

Consider the sequence
1, 5, 6, 25, 26, 30, 31, ...
which consists of every positive integer that can be expressed as a sum of distinct powers of 5.
What is the first term that is greater than 50?

Consider the sequence
1, 3, 4, 9, 10, 12, 13, ...
which consists of every positive integer that can be expressed as a sum of distinct powers of 3.
What is the first term that is greater than 20?

onyuIee Apr 19, 2025

0

8

0

+608

Counting

onyuIee Apr 19, 2025

0

12

0

+1808

Counting

Find the number of sequences containing three terms, such that
* The second term is equal to the sum of the first term plus one.
* The third term is equal to twice the second term.
* Each term is an integer in \{0, 1, 2, \dots, 100\} read more ..

blackpanther Apr 19, 2025

0

9

1

+56

Counting

Find the number of arithmetic sequences such that:
* The arithmetic sequence contains three terms
* All the terms are integers in \{0, 1, 2, \dots, 10\}
* The sum of the terms is 9.

Find the number of paths from A to B in the grid below, where each step is down or to the right.
The grid is 3 by 3, with A in the upper-left and B in the lower-right.

PikaPika989 Apr 18, 2025

0

12

0

+56

Counting

PikaPika989 Apr 18, 2025

0

5

1

+56

Counting

A basket contains 7 green marbles and 6 red marbles. Two marbles are taken from the basket at random without replacement (i.e., the first marble is not put back before the second marble is drawn). What is the probability that the first marble is green, read more ..

In the United States, there are 50 states. Each state is represented by 2 senators. In how many ways can we form a committee with 5 senators, in which no two of the senators are from the same state?

PikaPika989 Apr 18, 2025

0

14

0

+357

Counting

cwimie Apr 17, 2025

0

9

0

+357

Counting

Suppose for this problem (though it may not be accurate in real life) that the Senate has 48 Republicans and 52 Democrats. In how many ways can we form a committee with 5 senators, in which neither party holds all 5 seats?

cwimie Apr 17, 2025

0

12

1

+56

help with counting

Now suppose that not only must Sir Lancelot and Sir Gawain be next to each other, but Sir Galahad and Sir Percival also demand to be next to each other. How many seatings of the 8 knights are possible?

King Arthur's round table has 8 evenly spaced chairs. In how many ways can 8 knights be seated in the chairs, if Sir Lancelot and Sir Gawain insist on being seated next to each other?

PikaPika989 Apr 17, 2025

0

5

0

+56

Counting

PikaPika989 Apr 17, 2025

0

13

0

+56

Counting

In how many ways can 5 balls be placed in 4 boxes, if 2 balls are white and 3 balls are black? (Balls of the same color are indistinguishable. The boxes are indistinguishable.)

PikaPika989 Apr 17, 2025

0

9

1

+56

help with counting

How many zeroes do we write when we write all the integers from 1 to 625 in base 4?

The range of the function is . Let . Find the range of the function .
Thanks!

PikaPika989 Apr 17, 2025

+1

9

3

+16

Help Please Intermediate Algebra

Given a regular octagon, in how many ways can we color one diagonal red and another diagonal blue so that the two colored diagonals intersect at an endpoint? Consider rotations and reflections distinct.

PikaPika999 Apr 17, 2025

0

6

1

+810

Counting

A survey conducted among 150 high school students revealed that

* 68 students like Math
* 85 students like English
* 55 students like History
* 20 students like both Math and English
* 15 students like both Math and read more ..

crimefightingvigiI Apr 17, 2025

0

1

13

0

+970

Counting

gnistory Apr 17, 2025

0

1

10

0

+970

Counting

If an ant crawls from one corner to the other corner of a 3 \times 5 rectangle, then it will cross through seven squares.
If the ant crawls from one corner to the other corner of a 15 \times 16 rectangle, then how many squares will read more ..

gnistory Apr 17, 2025

0

12

1

+22

Using Geo/Trig on an Algebra Problem

What is $xy+yz+xz$ if $x^2+xy+y^2=1$, $y^2+yz+z^2=2$, and $x^2+xz+z^2=3$?

Find the number of paths from A to B in the grid below, so that
* Each step is down or to the right.
* The path cannot pass through any point more than once.

An example path is shown.
The grid is 3 by 3, with read more ..

Xenosmilus Apr 17, 2025

0

3

6

0

+483

Counting

bIueb3rry Apr 17, 2025

0

1

12

0

+483

Counting

Find the number of ways of filling in the squares of a 3 \times 3 grid so that:
* Each square contains a 0 or a 1.
* The sum of the numbers in each row and each column is at most 1.
An example is shown below.

bIueb3rry Apr 17, 2025

0

1

10

0

+483

Counting

How many ordered pairs of positive integers (m,n) satisfy \text{lcm}[m,n] = 360 and \gcd(m,n) = 360?

bIueb3rry Apr 17, 2025

0

3

11

0

+483

Counting

A bag contains red and blue tiles. Each tile has a number from the set \{-1, 0, 1\} written on it. I want to arrange 7 of these tiles in a row, so that the numbers on any three consecutive tiles sum to 3. In how many ways can this be done, assuming that read more ..

bIueb3rry Apr 17, 2025

0

14

0

+400

Counting

The following cards are split into four piles at random, so that every pile contains the same number of cards. What is the probability that every pile contains an Ace?
The cards are:
Ace of clubs
Ace of diamonds read more ..

Stanry Apr 17, 2025

0

9

0

+400

Counting

How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

Stanry Apr 16, 2025

0

4

1

+400

Counting

How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?

How many 7-digit sequences have a digit that appears at least 6 times?
(For example, 3339333 and 0200000 are two such sequences. A sequence is allowed to begin with 0.)

Stanry Apr 16, 2025

0

8

1

+400

Counting

Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.

Stanry Apr 15, 2025

0

11

0

+400

Counting

Stanry Apr 15, 2025

0

12

0

+400

Counting

We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen). What is the probability that we chose a multiple of 5?

Stanry Apr 14, 2025

0

17

0

+400

Counting

How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4?

Stanry Apr 14, 2025

0

13

0

+400

Counting

A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5).
Find the number of permutations (x_1, x_2, \dots, x_8) read more ..

Stanry Apr 13, 2025

0

8

0

+400

Counting

Find the number of sequences (a_1, a_2, a_3, \dots, a_8) such that:
* a_i \in \{1, 2, 3, 4, 5, 6, 7, 8\} for all 1 \le i \le 8.
* Every number1, 2, 3, 4, 5, 6, 7, 8 appears at least once in the sequence.

Stanry Apr 13, 2025

0

15

0

+400

Counting

A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit.
You are dealt a hand of read more ..

Stanry Apr 13, 2025

0

7

1

+400

Counting

In the array below, in how many different ways can we start with the letter and move from letter to letter (horizontally, vertically, or diagonally), to spell the word "ARCS"?
A read more ..

How many squares in the plane have at least two points in the lattice below as vertices?

Stanry Apr 13, 2025

0

16

0

+400

Counting

Stanry Apr 13, 2025

0

3

4

2

+400

Counting

At a meeting, two scientists, two mathematicians, two historians, and two artists are to be seated around a circular table. In how many ways can they be seated so that all four pairs of people from the same discipline are seated together?

Triangle ABC has AB = 6. Let D lie on BC such that \overline{AD} bisects \angle BAC. If BD = 3 and CD = 5, what is AC?

Stanry Apr 12, 2025

+1

1

10

3

+1524

Geometry

CPhill ●●●

learnmgcat Apr 12, 2025

0

5

0

+1524

Geometry

Find an acute angle A such that \sin 4A = \sin A + \sin 2A. Express your answer in degrees.

learnmgcat Apr 12, 2025

0

2

6

0

+1524

Geometry

What is the distance between the points (\cos 37^{\circ}, \sin 37^{\circ}) and (\cos 127^{\circ}, \sin 127^{\circ})?

learnmgcat Apr 12, 2025

0

3

6

1

+1524

Geometry

Chris places an orange cone at his current location. Then, he faces west, walks 40 meters, turns 30^{\circ} to his right, and walks 20 meters. How far is Chris from the cone, in meters? Round your answer to the nearest whole number.
(You will need read more ..

In a basketball tournament, there are four teams, and each team plays against every other team exactly twice. (So each team plays six games. Also, each team is equally likely to win a game, and there are no ties.) Find the probability read more ..

learnmgcat Apr 12, 2025

0

3

1

+810

Counting

Five standard 6-sided dice are rolled, and the resulting numbers are multiplied together. What is the probability that the product is divisible by 12?

crimefightingvigiI Apr 12, 2025

0

4

0

+810

Counting

crimefightingvigiI Apr 12, 2025

0

13

0

+810

Counting

Five people check identical suitcases before boarding an airplane. At the baggage claim, each person takes one of the five suitcases at random. What is the probability that exactly one person ends up with the wrong suitcase?

crimefightingvigiI Apr 12, 2025

0

14

0

+810

Counting

Find the number of positive integers that are divisors of at least one of 6^{6}, 10^{10}, 15^{15}, and 30^{30}.

crimefightingvigiI Apr 12, 2025

0

11

0

+810

Counting

I can only remember a seven-digit telephone number if the first three digits (the "prefix") are equal to the next three digits or the last three digits. For example, I can remember 389-3892 and 274-9274.
How many seven-digit telephone numbers can read more ..

crimefightingvigiI Apr 12, 2025

0

12

0

+1808

Counting

In how many ways can we seat 3 pairs of siblings in a row of 10 chairs, so that nobody sits next to their sibling? (Two chairs will be left empty, of course.)

blackpanther Apr 12, 2025

0

4

0

+1808

Counting

A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3)$ is a permutation of (1,2,3,4,5).
Let \pi = (x_1,x_2,x_3,\dots,x_n) be a permutation read more ..

blackpanther Apr 12, 2025

0

11

0

+1808

Counting

If an ant crawls from one corner to the other corner of a 3 \times 5 rectangle, then it will cross through seven squares.
If the ant crawls from one corner to the other corner of a 15 \times 16 rectangle, then how many squares will read more ..

blackpanther Apr 12, 2025

0

6

0

+1808

Counting

A survey conducted among 150 high school students revealed that

* 68 students like Math
* 85 students like English
* 55 students like History
* 20 students like both Math and English
* 15 students like both Math and read more ..

blackpanther Apr 11, 2025

+1

7

1

+608

Counting

Let P be a point chosen uniformly at random inside ABC. Extend ray BP to hit side AC at D. What is the probability that BD < 4?
The sides of triangle ABC are 3, 5, and 7.

Given a regular octagon, in how many ways can we color one diagonal red and another diagonal blue so that the two colored diagonals intersect at an endpoint? Consider rotations and reflections distinct.

onyuIee Apr 10, 2025

0

15

0

+608

Counting

onyuIee Apr 10, 2025

0

9

0

+608

Counting

Find the number of ways of placing three As, three Bs, and three Cs in a 3 \times 3 grid, so that every square contains one letter, and each diagonals contains one A, one B, and one C.

onyuIee Apr 10, 2025

0

13

1

+190

Counting

How many zeroes do we write when we write all the integers from 1 to 625 in base 4?

In how many ways can 5 balls be placed in 4 boxes, if 2$balls are white and 3 balls are black? (Balls of the same color are indistinguishable. The boxes are indistinguishable.)

HumanBemg Apr 9, 2025

0

12

1

+190

Counting

Now suppose that not only must Sir Lancelot and Sir Gawain be next to each other, but Sir Galahad and Sir Percival also demand to be next to each other. How many seatings of the 8 knights are possible?

HumanBemg Apr 9, 2025

0

14

0

+190

Counting

HumanBemg Apr 8, 2025

0

9

0

+190

Counting

King Arthur's round table has 8 evenly spaced chairs. In how many ways can 8 knights be seated in the chairs, if Sir Lancelot and Sir Gawain insist on being seated next to each other?

HumanBemg Apr 8, 2025

0

1

8

0

+943

Counting

Suppose for this problem (though it may not be accurate in real life) that the Senate has 48 Republicans and 52 Democrats. In how many ways can we form a committee with 5 senators, in which neither party holds all 5 seats?

booboo44 Apr 8, 2025

0

2

8

0

+943

Counting

In the United States, there are 50 states. Each state is represented by 2 senators. In how many ways can we form a committee with 5 senators, in which no two of the senators are from the same state?

booboo44 Apr 8, 2025

0

1

9

1

+943

Counting

A basket contains 7 green marbles and 6 red marbles. Two marbles are taken from the basket at random without replacement (i.e., the first marble is not put back before the second marble is drawn). What is the probability that the first marble is green, read more ..

A basket contains 7 green marbles and 6 red marbles. A marble is taken from the basket at random; its color is recorded, then the marble is returned to the basket. A second marble is then taken from the basket at random, and its color is recorded. read more ..

booboo44 Apr 8, 2025

0

1

11

1

+943

Counting

Let a_1, a_2, a_3, \dots be an infinite geometric series with positive terms. If a_2 = 10, then find the smallest possible value of
a_1 + a_2 + a_3.

booboo44 Apr 8, 2025

0

1

3

1

+970

Algebra

Compute
1.11111 + 0.11111 + 0.01111 + 0.00111 + 0.00011 + 0.00001

gnistory Apr 7, 2025

+1

6

2

+970

Algebra

Tackling Friction in Free body diagrams, would appreciate your generous help.
Problem: A block with a mass of 10 kg is placed on a rough inclined plane that makes a 30° angle with the horizontal. The coefficient of static read more ..

gnistory Apr 7, 2025

0

2

4

1

+448

Free Body Diagram

A and B are two points on a sphere of radius 2. We know the space distance between A and B is 2, What is distance from A to B along the (minor) arc of a great circle?

HiylinLink Apr 7, 2025

+1

4

2

+400

Geometry

Bosco ●●

Stanry Apr 7, 2025

-1

5

2

+190

Geometry

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of triangle ABC?
A is at (0,0), B is at (1,0), and C is at (1,1).

In \triangle ABC, we have AB = AC = 4 and \angle BAC = 45^\circ. If M is the midpoint of BC, then find AM^2.

HumanBemg Apr 5, 2025

+1

11

1

+190

Geometry

What is the equation of the reflection of the line y = 2x + 1 across the line y = 5x - 3? Give your answer in slope-intercept form.

HumanBemg Apr 5, 2025

+1

3

1

+190

Geometry

In the diagram below, ABCD is a rectangle with AD=6 and B=10. Point M is the intersection of its diagonals and point E lies on \overline{AD} such that BE=2. Find CE^2. (Diagram not to scale)

HumanBemg Apr 5, 2025

-1

2

8

1

+970

Geometry

Let R=(a,b) be the image of rotating point P=(7,2) clockwise by 150^\circ degrees around Q=(12,-5). What is b?

gnistory Apr 5, 2025

+1

1

3

1

+970

Geometry

Let R be the image of rotating point P=(4,0) counterclockwise by 60^\circ$ degrees around Q=(12,-7). What is PR?

gnistory Apr 5, 2025

0

1

11

0

+970

Geometry

gnistory Apr 5, 2025

0

1

8

1

+970

Geometry

Point B is the reflection of point A=(2,-1) over the line y=1. Point C is the reflection of point B over the line x + 2y = a. If C=(-5,8), what is a?

The area of a cross section of a sphere is 64\% of the largest possible cross sectional area of the sphere. If the sphere has radius 1/2, what is the area of the cross section?

gnistory Apr 4, 2025

0

2

3

1

+970

Geometry

The base of right pyramid ABCDE is a rhombus with side 5. We also know that \triangle ABD \cong \triangle CBD and EA=BA=2. Find the volume of the pyramid.

gnistory Apr 4, 2025

0

1

4

1

+943

Geometry

In the diagram, \overline{AD} is an altitude, \overline{BE} is a median, and \overline{AD}, \overline{BE}, and \overline{CF} are concurrent. If AB = 11, AC = 12, and AD = 6, what is AF?

booboo44 Apr 4, 2025

0

15

1

+294

Geometry

In the diagram below, \overline{AD} and \overline{BE} are angle bisectors of \angle BAC and \angle ABC, respectively, and they intersect at T. We know that BC = 12, AC = 18 and BF = 3 \cdot AF. Find AB.

AUnVerifedTaxPayer Apr 4, 2025

0

18

1

+400

i need help

The left column contains pairs of triangles with different side-length and angle-measure congruences marked. Match each diagram in the left column with a congruence/similarity criterion in the right column that justifies why the two triangles are congruent/similar. read more ..

Stanry Apr 4, 2025

0

9

1

+400

help me plz

The left column contains pairs of triangles with different side-length and angle-measure congruences marked. Match each diagram in the left column with a congruence/similarity criterion in the right column that justifies why the two triangles are read more ..

Stanry Apr 4, 2025

0

10

2

+400

need help

A stick is broken at two points, chosen at random. If the length of the stick is 6 units, then what is the probability that all three resulting pieces are shorter than 5 units?
What I've tried so far:
I've treated the stick as read more ..

Stanry Apr 4, 2025

0

13

2

+71

Counting and Probability, Melody or CPhill help ;-;

In the diagram below, we know \tan \theta = \frac{3}{4}. Find the area of the triangle.
The triangle is right, and the hypotenuse is 80.

Starry Apr 4, 2025

0

1

10

1

+1524

Geometry

Find the value of \sin 40^\circ \cdot \cos 50^\circ \cdot \cos 40^\circ \cdot \sin 50^\circ.

learnmgcat Apr 4, 2025

+2

1

8

1

+1524

Geometry

Find sin B.
BD = 15, CD = 10, AC = 6

learnmgcat Apr 4, 2025

0

1

4

0

+483

Geometry

bIueb3rry Apr 4, 2025

0

10

0

+483

Geometry

Find AC in the diagram to the nearest integer.
(You will need to use a calculator. Make sure to use "Degree" instead of "Radian" as the angle unit when computing trigonometric functions.)
AB = 31, angle A = 90 degrees, read more ..

bIueb3rry Apr 4, 2025

0

13

1

+483

Geometry

In \triangle PQR, we have \angle P = 90^\circ, PQ=3, and QR = 5. Find \tan R.

For a positive integer n, let f(n) denote the integer that is closest to \sqrt[4]{n}. Find the integer m so that
\sum_{n = 1}^m f(n) = 100.

bIueb3rry Apr 4, 2025

0

9

1

+810

Algebra

For how many integer values of a does the equation
x^2 + ax + 12a = 5x + 8
have integer solutions for x?

crimefightingvigiI Apr 3, 2025

0

7

0

+810

Algebra

crimefightingvigiI Apr 2, 2025

0

6

0

+1733

Algebra

Let a and b be positive real numbers. Let
m = \min \left\{ a, \frac{1}{b}, b^2 + \frac{1}{a + 1} \right\}.
Find the largest possible value of m.

parmen Apr 1, 2025

0

17

0

+190

Algebra

The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies
xf(x) + f(1 - x)/x = x^3 + 3x^2 + 14x - 13
for all real x. Find f(x).

HumanBemg Apr 1, 2025

+2

11

1

+6

Hello, guys and gals.

안녕하세요, 여러분. 저는 여러분이 파이에 대해 배우도록 이 글을 찾아오겠습니다. 파이는 정당한 스러울 수 있는 동그라미입니다. 그럼요. 이 문자 그대로 무한한 숫자의 여행을 시작하면 됩니다. 그리고, 위의 경고를 꼭 읽어주세요. 피자는 날씨가 좋으니까요.
1. PI란 무엇인가?
파이는 원의 직경과 원둘레이의 경우입니다. 흥미로운 점은 우주만큼 크든 원자핵만큼 작든 모든 원에 대해 동일하다는 것입니다.
원의 read more ..

Find all complex solutions to the equation z^3 = 8i - (15 + 2i)z^2 + (7 - 4i)z.

xhxmsja Mar 31, 2025

0

19

1

+44

Algebra

Find all complex numbers z such that |z - 1| = |z + 3| + |z - 2i|.

XenosmiIus Mar 30, 2025

0

13

0

+44

Algebra

XenosmiIus Mar 30, 2025

0

15

1

+44

Algebra

For parts (a) through (d), let f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.
What is the vertex of the parabola y = f(x)?

What is the equation of the line of symmetry of the parabola $y = f(x)$.
f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.

XenosmiIus Mar 30, 2025

0

4

2

+44

Algebra

In quadrilateral $BCED$, sides $\overline{BD}$ and $\overline{CE}$ are extended past $B$ and $C$, respectively, to meet at point $A$. If $BD = 8$, $BC = 3$, $CE = 1$, $AC = 19$ and $AB = 13$, then what is $DE$?

XenosmiIus Mar 30, 2025

-1

1

7

3

+490

Geometry

In triangle $ABC$, points $D$ and $F$ are on $\overline{AB},$ and $E$ is on $\overline{AC}$ such that $\overline{DE}\parallel \overline{BC}$ and $\overline{EF}\parallel \overline{CD}$. If $CE =3$ and $DF = 3$, then what is $BD$?

MeIdHunter Mar 30, 2025

-1

2

1

+490

Geometry

In triangle $PQR,$ $M$ is the midpoint of $\overline{QR}.$ Find $PM.$
PQ = 5, PR = 8, QR = 11

MeIdHunter Mar 30, 2025

0

1

6

1

+490

Geometry

Find the area of triangle ABC
[asy]
unitsize (2 cm);
pair A, B, C, D, E, F, H;
A = (0,2);
B = (-1,0);<#> C = (1,0);<#> D = (0,0);<#> E = (B + reflect(A,C)*(B))/2;<#> F = (C + reflect(A,B)*(C))/2;<#> read more ..

MeIdHunter Mar 30, 2025

0

4

1

+4

help please

Let x_1, x_2, \dots, x_{100} be real numbers. If
x_1 + 2x_2 + \dots + 100x_{100} = 1,
then find the minimum value of x_1/1 + x_2/2 + \dots + x_{100}/100.

whatthedonut Mar 30, 2025

0

16

0

+44

Algebra

XenosmiIus Mar 30, 2025

0

18

0

+44

Algebra

Let x, y, and z be real numbers. If x^2 + y^2 + z^2 = 1, then find the maximum value of
3x + 4y + 5z + x^3 + \frac{4x^2 y}{z} + \frac{z^5}{xy^2}

XenosmiIus Mar 30, 2025

0

10

0

+44

Algebra

Let x_1, x_2, \dots, x_n be real numbers. If
x_1^2 + 2x_2^2 + \dots + nx_n^2 = 1,
then find the maximum value of (x_1 + x_2/2 + \dots + x_n/n)^2, in terms of n.

XenosmiIus Mar 30, 2025

0

18

0

+44

Help algebra help

Let x_1, x_2, \dots, x_n be real numbers. If
x_1 + 2x_2 + \dots + nx_n = 1,
then find the minimum value of x_1^2/1 + x_2^2/2 + \dots + x_n^2/n, in terms of n.

XenosmiIus Mar 30, 2025

0

12

0

+44

Help me algebra

Let a, b, and c be positive real numbers. Find the minimum value of
(a + 1)^2 + \left( \frac{b}{a} + a + 1 \right)^2 + \left( \frac{c}{b} + abc \right)^2 + \left( \frac{4}{c} + \frac{c}{a} \right)^2.

XenosmiIus Mar 30, 2025

0

14

0

+44

Algebra question

Let x and y be positive real numbers. If
\frac{x^2}{9} + \frac{y^2}{25} = 1,
then find the maximum value of \frac{x^2}{3} \cdot \frac{y^2}{5}.

XenosmiIus Mar 30, 2025

0

19

0

+44

Help with Algebra

Let a, b, and c be positive real numbers. If \frac{a}{2} + b + 3c = 12, then find the maximum value of
\min \left\{ \frac{ab}{2}, ac, 12bc \right\}.

XenosmiIus Mar 30, 2025

+1

14

3

+22

Interesting Range Question; Can use polynomial long division to solve

Find the range of the function

Enter your answer in interval notation.

Find all x such that
(2x)^{\log_{10} 2} = (9x)*{\log_{10} 9} + (5x)^{\log_{10} 9} + \log_x 243

Xenosmilus Mar 29, 2025

0

8

0

+190

Algebra

HumanBemg Mar 29, 2025

0

13

0

+190

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

HumanBemg Mar 29, 2025

0

11

0

+1808

Algebra

Suppose that f(x) and g(x) are functions which satisfy
f(g(x)) = x^2 \quad \text{and} \quad g(f(x)) = x^4
for all x \ge 1. If g(16) = 1, then compute \log_2 g(2).

blackpanther Mar 27, 2025

0

19

0

+190

Algebra

Suppose f(x) is a rational function such that
3 f \left( \frac{1}{x} \right) - \frac{f(x)}{x} = x
for all $x \neq 0$. Find f(-2).

HumanBemg Mar 27, 2025

0

15

0

+190

Algebra

Let F(x) be the real-valued function defined for all real x except for x = 1 and x = 2 and satisfying the functional equation
F(x) + F \left( \frac{2x - 3}{x - 1} \right) + F \left( \frac{1}{x} \right) = x.
Find the function F(x) satisfying read more ..

HumanBemg Mar 27, 2025

0

19

0

+190

Algebra

The function f(n) is defined for all integers n, such that
f(x) + f(y) = f(x + y) - 4xy - 1 + f(x^2) + f(y^2)
for all integers x and y, and f(1) = 1. Find f(n).

HumanBemg Mar 27, 2025

0

12

0

+190

Algebra

The function f(n) takes the integers to the real numbers such that
f(m + n) + f(m - n) = 2f(m) + 2f(n) + mn
for all integers m and n, and f(1) = 2. Find f(n).

HumanBemg Mar 27, 2025

0

16

1

+1808

Algebra

The function f : \mathbb{R} \rightarrow \mathbb{R} satisfies
f(x) f(y) - f(xy) = -2x - 6y + 10
for all x, y \in \mathbb{R}. Find f(x).

Let f be a function defined on the positive integers, such that
f(xy) = f(x) + f(y)
for all positive integers x and y. Given that f(5) = 6, f(65) = 7, f(86) = 9, f(93) = 10, find (120).

blackpanther Mar 26, 2025

0

14

2

+1808

Algebra

The function f has the following properties:
* f(a,b) is defined for all positive integers a and b
* f(a,1) = a
* f(a,b) = 1 if b > a
* f(a + 1,b) = b[f(a,b) - f(a,b - 1)] read more ..

blackpanther Mar 26, 2025

0

13

1

+1808

Algebra

Fill in the blanks, to make a true equation:
(8x^3 + 24x^2 + 15x + 1)/((x^2 - 1)(x^2 + 3x)) = ___/(x - 1) + ___/(x + 3) + ___/x + ___/(x + 1)

blackpanther Mar 25, 2025

0

14

0

+1808

Algebra

blackpanther Mar 25, 2025

0

16

0

+1808

Algebra

Find the all real numbers that are not in the domain of f(g(x)), where
f(x) = \frac{3x^2 - 10x - 25}{x + 1} and g(x) = \frac{14x - 6}{3x^2 + 5x + 15}

blackpanther Mar 25, 2025

0

17

0

+1808

Algebra

Give a polynomial g(x) so that f(x) + g(x) has a horizontal asymptote of y = 0 as x approaches positive infinity, where
f(x) = \frac{2x^4 - 3x^3 - 8x^2 + 4x - 4}{x^2 + x}.

blackpanther Mar 25, 2025

0

8

0

+1808

Algebra

Fill in the blanks, to make a true equation:
\frac{2x^4 - 3x^3 - x^2 + 4x - 4}{x^2 + x} = ___x^2 + ___x + ___ + ___/x + ___/(x + 1).

blackpanther Mar 25, 2025

0

11

0

+190

Algebra

Find the largest value of x where the plots of
f(x) = - \frac{2x + 5}{x + 3} and g(x) = \frac{12}{x - 1}
intersect.

HumanBemg Mar 25, 2025

0

15

0

+190

Algebra

Fill in the blanks with constants, to make a true equation:
\frac{x^2 - 6x - 3}{x^3 - 4x} = ___/x + ___/(x - 2) + ___/(x + 2)

HumanBemg Mar 25, 2025

0

18

0

+190

Algebra

Let (x,y,z) be the real solution to the system of equations
x + y = \sqrt{4z + 3}
y + z = \sqrt{4x - 1}
z + x = \sqrt{4y + 5}
Find x + y + z.

HumanBemg Mar 25, 2025

0

20

0

+190

Algebra

Find all x such that
sqrt{3x^2 + 2x + 1} + sqrt{3x^2 + 2x - 3} = 20.

HumanBemg Mar 25, 2025

0

12

1

+8

For part b, we solve the answer as 11. The answer given by the book is 44. May i know why is it 44?

Eddie has some white, blue and yellow marbles. He has 4/5 as many white marbles as blue marbles and 120% as many blue marbles as yellow marbles.
(a) Find the ratio of the number of white marbles to the number of blue marbles to the number read more ..

Simplify \frac{6 \sqrt{2]}{\sqrt{2} - \sqrt{3} - \sqrt{5}}

sPpBCdIT Mar 25, 2025

0

15

0

+343

Algebra

jaekg Mar 24, 2025

0

1

10

1

+343

Algebra

Fill in the blanks.
If f(x) is an even function and g(x) is an even function and h(x) is an even function, then f(x) + g(x) + h(x) is an ___ function.
If f(x) is an odd function and g(x) is an odd function and h(x) read more ..

Find the value of x that maximizes
f(x) = \log (-20x + 16 \sqrt{x} - x).

jaekg Mar 24, 2025

0

14

1

+343

Algebra

If 2^{16^x} = 128*4^x, then find 2^x.

jaekg Mar 24, 2025

0

17

0

+343

Algebra

jaekg Mar 24, 2025

0

17

2

+194

someone helpp

Find an expression for the number of nonnegative integer triples (x, y, z) which are solutions of the equation x + y + z = n, where n is a positive integer.
I believe it has something to do with the hockey stick identity, but read more ..

The function f has the following properties:
* f(x) is defined for x > 0
* f(x) > 0 for all x > 0
* f(x - y) = \sqrt{f(xy) + 4x + 4y + 8} for all x > y > 0

Determine f(1).

HumenBeing Mar 23, 2025

0

16

0

+810

Algebra

crimefightingvigiI Mar 22, 2025

0

12

0

+810

Algebra

Let f be a function such that
f(x) + f(2x + y) + 5xy = f(4x - y) - x^2 + 5xy - 8x + 17y + 1
for all real numbers x and y. Find f(10).

crimefightingvigiI Mar 22, 2025

0

14

0

+190

Algebra

Let f be a function such that
f(xy) + x = xf(y) + f(x) + xy^2
for all real numbers x and y. If f(-1) = 3, then compute f(100).

HumanBemg Mar 21, 2025

0

17

0

+190

Algebra

Let a, b, and c be positive real numbers. If a + b + c = 1, then find the minimum value of
\frac{1}{a} + \frac{1}{b} + \frac{4}{c*a^2} + \frac{16}{b^4*a} + \frac{32}{a*b^3}

HumanBemg Mar 21, 2025

0

21

0

+190

Algebra

Let a, b, and c be positive real numbers. If a + b + c = 1, then find the minimum value of
\frac{1}{a} + \frac{1}{b} + \frac{1}{c*a^2} + \frac{2}{ab^2} + \frac{8}{c^3}.

HumanBemg Mar 21, 2025

+1

7

3

+120

Algebra

The function f satisfies
f(m + n) = f(m) + f(n) - 2f(mn + m + n + 1) + m^2 + n^2
for all nonnegative integers m and n, and f(1) = 0. Compute f(123).

The function f satisfies
f(a + b) = f(a) + f(b) - ab
for all nonnegative integers a and b, and f(1) = 7. Compute f(123).

coIinqiu Mar 20, 2025

0

6

0

+1285

Algebra

AnswerscorrectIy Mar 20, 2025

0

13

0

+1285

Algebra

The real numbers x_1, x_2, x_3, x_4, and x_5 satisfy
x_1 + x_2 + x_3 + x_4 + x_5 = 8,
x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 12,
x_1^3 + x_2^3 + x_3^3 + x_4^3 + x_5^3 = 16.
Let m be the smallest possible value of read more ..

AnswerscorrectIy Mar 20, 2025

0

5

0

+1285

Algebra

(a) Let x, y, and z be positive real numbers. Find the largest possible value of
\sqrt{\frac{3x + 5y + 2z}{6x + 5y + 4z}} + \sqrt{\frac{2x + 5y + z}{6x + 5y + 5z}} + \sqrt{\frac{9x + y + 4z}{6x + 5y + 4z}}.
(b) Find read more ..

AnswerscorrectIy Mar 20, 2025

0

6

0

+1285

Algebra

Let x_1, x_2, \dots, x_{100} be real numbers. If
x_1^2 + 2x_2^2 + \dots + 100x_{100}^2 = 1,
then find the maximum value of x_1/1 + x_2/2 + \dots + x_{100}/100.

AnswerscorrectIy Mar 20, 2025

0

10

0

+1285

Algebra

Let x_1, x_2, \dots, x_{100} be real numbers. If
x_1 + 2x_2 + \dots + 100x_{100} = 1,
then find the minimum value of x_1/1 + x_2/2 + \dots + x_{100}/100.

AnswerscorrectIy Mar 20, 2025

0

13

0

+1285

Algebra

Let x, y, and z be real numbers. If x^2 + y^2 + z^2 = 1, then find the maximum value of
3x + 4y + 5z + x^3 + \frac{4x^2 y}{z} + \frac{z^5}{xy^2}

AnswerscorrectIy Mar 19, 2025

0

16

0

+1285

Algebra

Let x_1, x_2, \dots, x_n be real numbers. If
x_1^2 + 2x_2^2 + \dots + nx_n^2 = 1,
then find the maximum value of (x_1 + x_2/2 + \dots + x_n/n)^2, in terms of n.

AnswerscorrectIy Mar 19, 2025

0

14

0

+1285

Algebra

Let x_1, x_2, \dots, x_n be real numbers. If
x_1 + 2x_2 + \dots + nx_n = 1,
then find the minimum value of x_1^2/1 + x_2^2/2 + \dots + x_n^2/n, in terms of n.

AnswerscorrectIy Mar 19, 2025

0

10

0

+1285

Algebra

Let a, b, and c be positive real numbers. Find the minimum value of
(a + 1)^2 + \left( \frac{b}{a} + a + 1 \right)^2 + \left( \frac{c}{b} + abc \right)^2 + \left( \frac{4}{c} + \frac{c}{a} \right)^2.

AnswerscorrectIy Mar 19, 2025

0

13

0

+120

Algebra

Let x and y be positive real numbers. If
\frac{x^2}{9} + \frac{y^2}{25} = 1,
then find the maximum value of \frac{x^2}{3} \cdot \frac{y^2}{5}.

coIinqiu Mar 19, 2025

0

9

0

+120

Algebra

Let a, b, and c be positive real numbers. If \frac{a}{2} + b + 3c = 12, then find the maximum value of
\min \left\{ \frac{ab}{2}, ac, 12bc \right\}.

coIinqiu Mar 19, 2025

0

15

1

+1285

Algebra

Let x and y be positive real numbers. If x + y = 1, then find the maximum value of xy + y^3.

Find the minimum value of
\frac{x^2}{x + 2}
for x > 2.

AnswerscorrectIy Mar 18, 2025

0

6

1

+1285

Algebra

Find the minimum value of the product
P(x,y,z) = (2x + 3y)(x + 4z) \left( y + \frac{5}{3} z \right),
when xyz = 1, and x, y, z are positive real numbers.

AnswerscorrectIy Mar 18, 2025

0

19

0

+1285

Algebra

AnswerscorrectIy Mar 18, 2025

0

10

0

+120

Algebra

Find the maximum p such that
2x^4 y^2 + \frac{9}{4} y^4 z^2 + \frac{3}{4} z^4 x^2 + 3x^3 y^3 + 10x^3 z^3 + 15y^3 z^3 - px^2 y^2 z^2
is always nonnegative for all real x, y, and z.

coIinqiu Mar 17, 2025

0

14

0

+120

Inequality

Let w, x, y, and z be positive real numbers. If w + 2x + 3y + 6z = 8 - w^2 - x^2 - y^2 - z^2, then what is the maximum value of wxyz?

coIinqiu Mar 17, 2025

0

15

3

+141

Super Confusing Math Problem ( Read Description )

PLEASE READ DESCRIPTION BEFORE ANSWERING
I first picked option A because thats what I think was right and AI and even google was saying its correct... but it ended up being wrong so then I picked option C. because that was my next bet read more ..

Let a and b be positive real numbers. Find the minimum value of
(a + b) \left( \frac{1}{a + 1} + \frac{1}{b + 1} \right)

acyclics Mar 17, 2025

0

13

0

+1285

Algebra

AnswerscorrectIy Mar 17, 2025

0

12

0

+1285

Algebra

Fill in the blanks, to make a true equation.
3/(3^2 - 1) + 3^2/(3^4 - 1) + 3^3/(3^6 - 1) + 3^4/(3^8 - 1) + ... + 3^(2(n - 1))/(3^(2n) - 1) = ___/___
Hint: Let S_n = \frac{3}{3^2 - 1} + \frac{3^2}{3^4 - 1} + \frac{3^3}{3^6 - 1} read more ..

AnswerscorrectIy Mar 17, 2025

0

17

2

+1285

Algebra

Find the number of points of intersection between the graphs of the following equations: \begin{align*}
y = |2x + 5|,
y = -|3x - 2| + |4x - 7| + x.

Let x be a positive real number. Show that
\frac{1}{x} \ge 3 - 2x
Describe when we have equality.

AnswerscorrectIy Mar 16, 2025

0

9

3

+1285

Algebra

Let
f(x) = \frac{x^4 + 2x^3 + 3x^2 + 2x + 1}{x}.
Find the minimum value of x for x > 0.

AnswerscorrectIy Mar 16, 2025

0

11

1

+1285

Algebra

Let $x$ and $y$ be real numbers such that
x^2 + 3xy + 2y^2 = 1 - 12x + 5y.
Find all possible values of x + y. Enter your answer using interval notation.

AnswerscorrectIy Mar 16, 2025

0

10

0

+1285

Algebra

AnswerscorrectIy Mar 16, 2025

0

15

0

+1285

Algebra

Let a and b be real numbers such that
(a^2 + 1)(b^2 + 4) = 14ab + 21.
Find the largest possible value of a^2 + b^2.

AnswerscorrectIy Mar 16, 2025

0

11

0

+1285

Algebra

Let x and y be real numbers satisfying
\frac{x^2y^2 - 1}{2y - 1} = 4x + y.
Find the largest possible value of x.

AnswerscorrectIy Mar 16, 2025

0

16

0

+1285

Algebra

Let x and y be real numbers. Find the maximum value of $(x + y)^2$, if $x$ and $y$ satisfy $x^2 + y^2 = 5 + 2xy$.

AnswerscorrectIy Mar 16, 2025

0

23

0

+1285

Algebra

Find the maximum value of f(a), where a is a positive integer, and
f(x) = \frac{13 + x}{2x + 7 - x^2}.

AnswerscorrectIy Mar 16, 2025

0

13

0

+1285

Algebra

Compute
\sum_{k = 1}^{100} k (\lceil \log_3 k \rceil - \lfloor \log_9 k \rfloor).

AnswerscorrectIy Mar 16, 2025

0

18

0

+1285

Algebra

Find the smallest positive real number x such that
\lfloor x^2 \rfloor - x \lfloor x \rfloor = \lfloor x^3 \rlfoor/x

AnswerscorrectIy Mar 16, 2025

0

12

0

+1285

Algebra

Find all x for which
\Big| x - |x - 2| \Big| = \lfloor x \rfloor + \lfloor 2x \rfloor
Express your answer in interval notation.

AnswerscorrectIy Mar 16, 2025

0

19

0

+1285

Algebra

Find all real numbers x that satisfy the equation
|x + 4| + |x - 7| = 3x - 1 + |3x - 7|.
If you find more than one such value of x, list all of your solutions separated by commas. If you only find one solution, then just enter that so read more ..

AnswerscorrectIy Mar 16, 2025

0

18

1

+1285

Algebra

Compute
\frac{\{\sqrt{3}\} - 4 \{\sqrt{5}\}}{\{\sqrt{3}\}^2 + \{\sqrt{2}\}^2}.

Find the area of the region determined by the system
y \ge |x|, \\
y \le -|2x + 1| + 6.

AnswerscorrectIy Mar 16, 2025

0

21

0

+1285

Algebra

AnswerscorrectIy Mar 15, 2025

0

16

0

+1285

Algebra

Let f be the piecewise function such that
f(x) =
x^2 - 5x - 64 & \text{if} \ x \le 0, \\
x^2 + 3x - 38 & \text{if} \ x > 0.
Find all x such that f(x) = 50.

AnswerscorrectIy Mar 15, 2025

0

27

0

+1285

Algebra

Find the solutions to
|x^2 + 8x| = 9 - |x^4 + 3x^2 + 2|

AnswerscorrectIy Mar 15, 2025

0

13

0

+810

Algebra

What is the horizontal asymptote as x approaches positive infinity of the graph of
y = \sqrt{4x^2 + 5x} - \sqrt{4x^2}?
The horizontal asymptote is in the form y = mx + k.

crimefightingvigiI Mar 15, 2025

+1

12

2

+810

Algebra

Find all real x where
2 \cdot \frac{x - 5}{x - 3} > \frac{2x - 5}{x + 2} + 20.
Give your answer in interval notation.

Find the values of x where the vertical asymptotes of f(g(x)) are located, where
f(x) = \frac{2x - 8}{x^2 - 2x - 3} and g(x) = \frac{x^3 + 2x + 9}{x^2 + 4}.

crimefightingvigiI Mar 15, 2025

0

14

0

+810

Algebra

crimefightingvigiI Mar 15, 2025

0

11

1

+120

Algebra

Fill in the blanks, to make a true equation:
(8x^3 + 24x^2 + 15x + 1)/((x^2 - 1)(x^2 + 3x)) = ___/(x - 1) + ___/(x + 3) + ___/x + ___/(x + 1)

Find the all real numbers that are not in the domain of f(g(x)), where
f(x) = \frac{3x^2 - 10x - 25}{x + 1} and g(x) = \frac{14x - 6}{3x^2 + 5x + 15}

coIinqiu Mar 14, 2025

0

15

0

+120

Algebra

coIinqiu Mar 14, 2025

0

15

0

+120

Algebra

Give a polynomial g(x) so that f(x) + g(x) has a horizontal asymptote of y = 0 as x approaches positive infinity, where
f(x) = \frac{2x^4 - 3x^3 - 8x^2 + 4x - 4}{x^2 + x}.

coIinqiu Mar 14, 2025

0

15

0

+120

Algebra

Fill in the blanks, to make a true equation:
\frac{2x^4 - 3x^3 - x^2 + 4x - 4}{x^2 + x} = ___x^2 + ___x + ___ + ___/x + ___/(x + 1).

coIinqiu Mar 14, 2025

0

18

0

+120

Algebra

Find the largest value of x where the plots of
f(x) = - \frac{2x + 5}{x + 3} and g(x) = \frac{12}{x - 1}
intersect.

coIinqiu Mar 14, 2025

0

10

2

+120

Algebra

Fill in the blanks with constants, to make a true equation:
\frac{x^2 - 6x - 3}{x^3 - 4x} = ___/x + ___/(x - 2) + ___/(x + 2)

Let (x,y,z) be the real solution to the system of equations
x + y = \sqrt{4z + 3}
y + z = \sqrt{4x - 1}
z + x = \sqrt{4y + 5}
Find x + y + z.

coIinqiu Mar 13, 2025

0

12

1

+120

Algebra

Find all x such that
sqrt{3x^2 + 2x + 1} + sqrt{3x^2 + 2x - 3} = 20.

coIinqiu Mar 13, 2025

0

12

0

+120

Algebra

coIinqiu Mar 13, 2025

0

11

0

+120

Algebra

Simplify \frac{6 \sqrt{2]}{\sqrt{2} - \sqrt{3} - \sqrt{5}}

coIinqiu Mar 13, 2025

0

15

0

+120

Algebra

Fill in the blanks.
If f(x) is an even function and g(x) is an even function and h(x) is an even function, then f(x) + g(x) + h(x) is an ___ function.
If f(x) is an odd function and g(x) is an odd function and h(x) read more ..

coIinqiu Mar 12, 2025

0

8

1

+120

Algebra

Find the value of x that maximizes
f(x) = \log (-20x + 16 \sqrt{x} - x).

If 2^{16^x} = 128*4^x, then find 2^x.

coIinqiu Mar 12, 2025

0

5

1

+120

Algebra

Express 2 + sqrt(2) + 1/(2 + sqrt(3)) + 1/(sqrt(5) - 2) in simplest form.

coIinqiu Mar 12, 2025

0

5

1

+120

Algebra

For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = x + 2 \lfloor x \rfloor + sqrt{x} - \frac{1}{|x|}

coIinqiu Mar 12, 2025

0

17

0

+1285

Algebra

AnswerscorrectIy Mar 11, 2025

0

16

0

+1285

Algebra

For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = \frac{x}{\sqrt{x^2 + 1}} + \frac{1}{x^2} - \frac{x}{x^2 - 1 read more ..

AnswerscorrectIy Mar 11, 2025

0

19

0

+1285

Algebra

For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = \frac{\sqrt{|x|}}{x} + x + \frac{1}{x^2}

AnswerscorrectIy Mar 11, 2025

0

13

0

+1285

Algebra

For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = \frac{1}{\sqrt{x^2 + 1}} - \frac{1}{x}

AnswerscorrectIy Mar 11, 2025

+1

1

11

2

+120

Algebra

For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = sqrt{10 - x} + \sqrt{x + 10}

For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = x^3 - 3x^2 + 10x - 15

coIinqiu Mar 11, 2025

+1

1

8

1

+120

Algebra

Find all real numbers a and b such that
a + b = 14
a^3 + b^3 = 812 + a^2 + b^2

coIinqiu Mar 11, 2025

0

9

2

+120

Algebra

Factor (ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3 + (a^2 + b^2 + c^2)^3 - 3(a^3 + b^3 + c^3)^2 as much as possible.

coIinqiu Mar 11, 2025

+1

17

0

+120

Algebra

coIinqiu Mar 11, 2025

0

25

0

+120

Algebra

Let P = \log_8 3 and Q = \log_3 5. Express \log_{15} 72 in terms of P and Q. Your answer should no longer include any logarithms.

coIinqiu Mar 11, 2025

0

13

0

+810

Algebra

Let x, y, and z be positive real numbers with x > y > z > 1 such that
\log_x y + \log_y z + \log_z x = \frac{13}{2},
\log_x z + \log_z y = 8.

Find \log_z x.

crimefightingvigiI Mar 11, 2025

0

9

1

+810

Algebra

The equation
\sqrt[5]{7} x^{\log_7 x} = x^{\log_5 x}
has two positive roots a and b. Compute ab.

Find \frac{a}{b} when 2 log(a - 2b) = log a + log (2b) - log(a + b).

crimefightingvigiI Mar 11, 2025

0

19

1

+1285

Algebra

If the two numbers \log_x 3 + \log_x 9 and \log_x 27 are equal, then what is their common value?

AnswerscorrectIy Mar 10, 2025

0

14

1

+1285

Algebra

Given that log_{4n} 40 sqrt(3) = log_{2n} 10, find n^4.

AnswerscorrectIy Mar 10, 2025

0

13

1

+1285

Algebra

Let x, y, and z all exceed 1, and let w be a positive number such that \log_x w = 24, \log_x yx = 40, and \log_{xy^2} zw = 12. Find \log_z w.

AnswerscorrectIy Mar 10, 2025

0

14

1

+810

Algebra

crimefightingvigiI Mar 10, 2025

+1

16

1

+7

Two squares are inscribed in a semi-circle, as shown below. Find the radius of the semi-circle.

Let P = \log_8 3 and Q = \log_3 5. Express \log_{15} 72 in terms of P and Q. Your answer should no longer include any logarithms.

Tigergaming2464 Mar 9, 2025

0

9

1

+810

Algebra

Factor (ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3 + (a^2 + b^2 + c^2)^3 - 3(a^3 + b^3 + c^3)^2 as much as possible.

crimefightingvigiI Mar 8, 2025

0

22

0

+810

Algebra

crimefightingvigiI Mar 8, 2025

0

5

2

+810

Algebra

Find all real numbers a and b such that
a + b = 14
a^3 + b^3 = 812 + a^2 + b^2

Find x if \log_2 x^2 + \log_{1/2} x + 3 \log_4 x = 5.

crimefightingvigiI Mar 8, 2025

0

18

0

+190

Algebra

HumanBemg Mar 8, 2025

0

18

0

+190

Algebra

Simplify 25^{\frac{1}{2} - \log 5 + \sqrt{3}}.

HumanBemg Mar 8, 2025

0

1

8

1

+190

Algebra

Compute
\log_4 5 + \log_5 6 + \log_6 7+ \log_{2047} 2048 + \log_{2048} 2049

Fill in the blanks, to make a true equation.
3/(3^2 - 1) + 3^2/(3^4 - 1) + 3^3/(3^6 - 1) + 3^4/(3^8 - 1) + ... + 3^(2(n - 1))/(3^(2n) - 1) = ___/___

HumanBemg Mar 8, 2025

0

22

0

+190

Algebra

HumanBemg Mar 8, 2025

0

19

1

+190

Counting

Let S be the set {1, 2, 3, \dots, 10, 11, 12}. How many subsets of the set S have no two consecutive primes as members?

Let
A_0 = 0
A_1 = 1
A_n = A_{n - 1} + A_{n - 2} for n ge 2
There is a unique ordered pair (c,d) such that c \alpha^n + d \beta^n is the closed form for sequence A_n. Find the ordered pair (c,d).

HumanBemg Mar 7, 2025

0

13

0

+190

Algebra

HumanBemg Mar 7, 2025

0

13

0

+190

Algebra

Find the ordered pair (p,q) such that
F_n = p \alpha^n + q \beta^n.

HumanBemg Mar 7, 2025

0

11

1

+190

Algebra

The Fibonacci sequence, is defined by F_0 = 0, F_1 = 1, and F_n = F_{n - 2} + F_{n - 1}. It turns out that
F_n = \frac{\alpha^n - \beta^n}{\sqrt{5}},
where \alpha = \frac{1 + \sqrt{5}}{2} and \beta = \frac{1 - \sqrt{5}}{2}.
The read more ..

Find a closed form for
S_n = 1! \cdot (1^2 + 1) + 2! \cdot (2^2 + 2) + \dots + n! \cdot (n^2 + n).\]
for any integer n \ge 1. Your response should have a factorial.

HumanBemg Mar 7, 2025

0

15

1

+190

Algebra

For a positive integer k, let
S_k = 1 \cdot 1! \cdot 2 + 2 \cdot 2! \cdot 3 + \dots + k \cdot k! \cdot (k + 1).
Find a closed form for S_k.

HumanBemg Mar 7, 2025

0

11

0

+190

Algebra

HumanBemg Mar 7, 2025

0

25

0

+21

Algebra

Find
\sum_{k = 1}^{20} k(k^2 - 10k - 20)(k^2 + 1)

co1inqiu Mar 7, 2025

0

16

1

+810

Algebra

Compute
1.11111 + 0.11111 + 0.01111 + 0.00111 + 0.00011 + 0.00001

Let a_1, a_2, a_3, \dots be an infinite geometric series with positive terms. If a_2 = 10, then find the smallest possible value of
a_1 + a_2 + a_3.

crimefightingvigiI Mar 7, 2025

0

19

1

+810

Algebra

Let
a + ar + ar^2 + ar^3 + \dotsb
be an infinite geometric series. The sum of the series is 9. The sum of the cubes of all the terms is 36. Find the common ratio.

crimefightingvigiI Mar 7, 2025

0

10

0

+810

Algebra

crimefightingvigiI Mar 7, 2025

0

18

1

+810

Algebra

Let A = x^4 + x^3 + x^2 + x + 1 and B = x^4 - x^3 + x^2 - x + 1. Simplify A + B.

The closed form sum of
12 \left[ 1 \cdot 2^2 \cdot 3 + 2 \cdot 3^2 \cdot 4 + \dots + n (n + 1)^2 (n + 2) \right]
for n \ge 1 is n(n + 1)(n + 2) p(n) for some polynomial p(n). Find p(n).

crimefightingvigiI Mar 7, 2025

0

15

0

+1285

Algebra

AnswerscorrectIy Mar 7, 2025

0

13

0

+1285

Algebra

The sum
6 (1 \cdot 3 \cdot 5 + 2 \cdot 4 \cdot 6 + \dots + n(n + 2)(n + 4))
is equal to a polynomial f(n) for all n \ge 1.
Write f(n) as a polynomial with terms in descending order of n.

AnswerscorrectIy Mar 7, 2025

0

3

1

+1285

Algebra

A sequence a_1, a_2, a_3, \dots of positive integers has the following properties:
* The first three terms are in geometric progression.
* The second, third, and fourth terms are in arithmetic progression.
* In general, for all $i\ge1$, read more ..

A sequence of real numbers (a_n) is defined as follows: a_0 = 1, a_2 = 2, and
a_{n + 2} = \frac{a_{n + 1}}{a_n}
for n = 0, 1, 2, \dots. Find a_0 + a_1 + a_2 + \dots + a_{100}.

AnswerscorrectIy Mar 7, 2025

0

21

0

+1285

Algebra

AnswerscorrectIy Mar 6, 2025

0

7

1

+1285

Algebra

Let
P = 3^{1/3} \cdot 9^{1/9} \cdot 27^{1/27} \cdot 81^{1/81}.
Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possible value of $a + b.$

Let
P = 2^{1/2} \cdot 4^{1/4} \cdot 8^{1/8} \cdot 16^{1/16}
Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possible value of $a + b.$

AnswerscorrectIy Mar 6, 2025

0

21

1

+1285

Algebra

Let r be a real number such that |r| < 1. Express
\sum_{n = 0}^{\infty} n*r^n*(n + 1)*(n + 2)
in terms of r.

AnswerscorrectIy Mar 6, 2025

0

1

21

1

+190

Algebra

Find
\sum_{k = 0}^{10} (k + 3) \cdot 2^k \cdot (k - 3)

HumanBemg Mar 6, 2025

0

1

13

1

+190

Algebra

Simplify \frac{1 + 3 + 5 + ... + 1999 + 2001 + 2003}{2 + 4 + 6 + ... + 2000 + 2002 + 2004 + 2006 + 2008 + 2010}.

HumanBemg Mar 6, 2025

0

13

2

+190

Algebra

Find the sum
\frac{1}{7} + \frac{2}{7^2} + \frac{3}{7^3} + \frac{1}{7^4} + \frac{2}{7^5} + \frac{3}{7^6}

HumanBemg Mar 6, 2025

+1

1

11

1

+190

Algebra

Let a_1, a_2, a_3, \dots be an infinite geometric series with positive terms. If a_2 = 10, then find the smallest possible value of
a_1 + a_2 + a_3.

HumanBemg Mar 6, 2025

0

1

6

1

+120

Algebra

Let
a + ar + ar^2 + ar^3 + \dotsb
be an infinite geometric series. The sum of the series is 9. The sum of the cubes of all the terms is 36. Find the common ratio.

coIinqiu Mar 5, 2025

0

15

1

+120

Algebra

Let A = x^4 + x^3 + x^2 + x + 1 and B = x^4 - x^3 + x^2 - x + 1. Simplify A + B.

coIinqiu Mar 5, 2025

0

1

13

1

+120

Algebra

thatgirlaj ●

coIinqiu Mar 5, 2025

0

26

1

+120

Algebra

Compute
1.11111 + 0.11111 + 0.01111 + 0.00111 + 0.00011 + 0.00001

Simplify (x^4 + x^3 + x^2 + x + 1) + (x^4 - x^3 + x^2 - x + 1).

coIinqiu Mar 4, 2025

0

11

1

+120

Algebra

Find all real numbers a such that the roots of the polynomial
x^3 - 3x^2 + 17x + a
form an arithmetic progression and are not all real.

coIinqiu Mar 4, 2025

0

25

0

+810

Algebra

crimefightingvigiI Mar 4, 2025

0

23

0

+810

Algebra

Find the sum of the series
1 + \frac{1}{2} + \frac{1}{10} + \frac{1}{30} + \frac{1}{60} + ,..
where we alternately multiply by $\frac 12$ and $\frac 15$ and $\frac{1}{3}$ to get successive terms.

crimefightingvigiI Mar 4, 2025

0

16

1

+810

Algebra

The sum of the first three terms of a geometric sequence of integers is equal to seven times the first term, and the sum of the first four terms is $30$. What is the first term of the sequence?

Suppose the domain of f is (-1,3). Define the function g by
g(x) = f(2/x + x/2).
What is the domain of g?

crimefightingvigiI Mar 4, 2025

0

24

0

+810

Algebra

crimefightingvigiI Mar 4, 2025

0

25

0

+810

Algebra

Suppose the domain of f is (-1,3). Define the function g by
g(x) = f(x - 4x^2)
What is the domain of g?

crimefightingvigiI Mar 4, 2025

0

20

0

+810

Algebra

Suppose the domain of f is (-1,3). Define the function g by
g(x) = 5 - f(x) + f(5/x).
What is the domain of g?

crimefightingvigiI Mar 4, 2025

0

21

0

+810

Algebra

Suppose the domain of f is (-1,3). Define the function g by
g(x) = f((x + 1)(x - 2)).
What is the domain of g?

crimefightingvigiI Mar 4, 2025

0

30

0

+21

Algebra

Let t be a root of f(x) = x^3 - x + 3. Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.

co1inqiu Mar 4, 2025

+1

19

1

+21

Algebra

Let r, s, and t be solutions of the equation x^3 - 4x^2 - 7x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}

Let r, s, and t be solutions of the equation x^3 + 2x^2 - 5x + 15 = 0. Compute
\frac{1}{r - 2s - 2t} + \frac{1}{s - 2r - 2t} + \frac{1}{t - 2r - 2s}

co1inqiu Mar 4, 2025

0

26

0

+21

Algebra

co1inqiu Mar 4, 2025

0

17

1

+21

Algebra

Suppose r and s are the values of x that satisfy the equation
x^2 - 2mx + (m^2 - 6m + 11) = 0
for some real number m. Find the minimum real value of (r - s)^2.

Let be a complex number so that
Find z, then calculate .

co1inqiu Mar 4, 2025

+1

20

2

+11

Complex numbers... Help needed!

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 7n - 3mn + 15m + 8n + ___

Mathematix1981 Mar 3, 2025

0

15

1

+190

Algebra

Fill in the blanks, to complete the factorizaion:
(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)

HumanBemg Mar 3, 2025

0

21

0

+190

Algebra

HumanBemg Mar 3, 2025

0

27

0

+190

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

HumanBemg Mar 3, 2025

0

27

0

+190

Algebra

Factor 3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.

HumanBemg Mar 3, 2025

0

22

0

+1808

Algebra

Let a_1, a_2, a_3, \dots, a_{10}, a_{11}, a_{12} be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} = 0$ and $a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} = 0$, then find $a_1$.

blackpanther Mar 3, 2025

0

19

1

+1808

Algebra

The system of equations
\frac{xy}{x + y} = 1, \quad \frac{xz}{x + z} = 1, \quad \frac{yz}{y + z} = 2
has exactly one solution. What is $z$ in this solution?

Find the number of ordered pairs $(a,b)$ of integers such that
\frac{a + 2}{a + 1} = \frac{b}{3}.

blackpanther Mar 3, 2025

0

14

1

+1808

Algebra

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 2 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator read more ..

blackpanther Mar 3, 2025

0

10

1

+1808

Algebra

Evaluate the infinite geometric series
0.4 + 0.036 + 0.0000324 + ...
Express your answer as a fraction with integer numerator and denominator.

blackpanther Mar 3, 2025

0

13

1

+977

Algebra

The equation of the line below can be expressed in the form
\frac{x}{a} + \frac{y}{b} = 1.
Find a.
The lines passes through (5,8) and (-4,17).

magenta Mar 3, 2025

0

11

1

+977

Algebra

Laverne starts counting out loud by 5's. She starts with 2. As Laverne counts, Shirley sums the numbers Laverne says. When the sum finally exceeds 20, Shirley runs screaming from the room. What number does Laverne say that sends read more ..

magenta Mar 3, 2025

+1

6

1

+977

Algebra

Triangle ABC is a 30-60-90 right triangle with right angle at C, angle ABC = 60 degrees, and hypotenuse of length 2. Let P be a point chosen at random inside ABC, and extend ray BP to hit side AC at D. What is the probability that BD < √2?
<read more ..

magenta Mar 3, 2025

0

12

1

+194

New question :) someone help me please T_T

Guys, look at these guidelines: https://web2.0calc.com/questions/some-guidlines-for-question-askers
I think everyone on this site who asks a question should in fact show what they have tried. This is not a website to magically read more ..

HumenBeing Mar 3, 2025

+1

22

0

+37

Question Askers

Kromy Mar 2, 2025

+1

15

2

+810

Algebra

Evaluate
1 + \frac{i}{3} - \frac{1}{9} - \frac{i}{27} + \frac{1}{81},
where $i$ is the imaginary unit.

Evaluate
2 + \frac{6}{7} +\frac {18}{49} + \frac{54}{343} + 1 + \frac{2}{14} + \frac{3}{98}.

crimefightingvigiI Mar 2, 2025

0

17

2

+810

Algebra

Evaluate \sum_{n = 1}^{33} (5n + 2 - 4n + n^2 + 17).

crimefightingvigiI Mar 2, 2025

+1

13

1

+810

Algebra

Evaluate 21 + 27 + 33 + ... + 267 + ... + 6021.

crimefightingvigiI Mar 2, 2025

0

10

2

+810

Algebra

Let be real numbers such that
Find the maximum value of in terms of
I originally got the answer of 1 but it was wrong! Thanks for helping!

crimefightingvigiI Mar 2, 2025

0

12

1

+36

Algebra

Thanks for helping!
Let be real numbers such that
and is minimized. Find

crimefightingvigil Mar 2, 2025

0

11

1

+36

HELP Pls

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$

crimefightingvigil Mar 2, 2025

0

6

2

+972

Algebra

Laverne starts counting out loud by 5's. She starts with 2. As Laverne counts, Shirley sums the numbers Laverne says. When the sum finally exceeds 20, Shirley runs screaming from the room. What number does Laverne say that sends read more ..

Rangcr897 Mar 2, 2025

0

11

1

+972

Algebra

Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

Rangcr897 Mar 2, 2025

0

1

10

1

+1524

Algebra

Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)

learnmgcat Mar 2, 2025

0

13

1

+1556

Algebra

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $AC = 15$, find $BZ$.

kittykat Mar 2, 2025

0

12

1

+810

Geometry

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 12$, $AY = 12$, and $AC = 15$, find $BC$.

crimefightingvigiI Mar 2, 2025

0

21

1

+810

Geometry

I have $4$ different mathematics textbooks and $3$ different psychology textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row, if there must be a mathematics textbook exactly in the middle, and all the psychology textbooks are read more ..

crimefightingvigiI Mar 2, 2025

0

18

2

+343

help with counting

Each face of a cube is painted randomly one of the colors red, orange, yellow, green, blue or purple. What is the probability that the cube has at least one pair of faces that share an edge and are the same color? Express your answer as a decimal to the read more ..

jaekg Mar 2, 2025

0

2

+34

Counting and Probability

Let f(x) be a polynomial. Find the remainder when f(x) is divided by x(x - 1)(x - 2), if f(0) = 0, f(1) = 1, and f(2) = 2.

colinqiu Mar 2, 2025

0

6

1

+1808

Algebra

Let f(x) be a polynomial. Find the remainder when f(x) is divided by x(x - 1)(x - 2), if f(0) = 0, f(1) = 1, and f(2) = 4.

blackpanther Mar 2, 2025

0

15

1

+1808

Algebra

Let a, b, c be the roots of p(x) = x^3 + 7x^2 + 10x - 13 - 5x^3 + 25x^2 + 44x - 15. Find the value of
1/(ab + ac) + 1/(ab + bc) + 1/(ac + bc)

blackpanther Mar 2, 2025

0

20

1

+1808

Algebra

Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 0, what is b?

blackpanther Mar 2, 2025

+1

1

10

1

+417

Algebra

Find all (real or nonreal) x satisfying
(x - 3)^4 + (x - 5)^4 = -8 + 6(x - 3)(x - 5)^3 - 11(x - 3)^3 (x - 5).

nathanl6656 Mar 2, 2025

0

1

14

1

+417

Algebra

Express x^8 + x^4 y^4 - x^6 y^2 + x^3 y^5 - 4xy^7 - y^8 as the product of two polynomials of degree 2, and a polynomial of degree 4.

nathanl6656 Mar 2, 2025

0

13

1

+417

Algebra

Find the constant k so that
2x^2 + 5xy - 8y^2 + 7x + 25y + k
can be expressed as the product of two linear factors of the form ax + by + c.

nathanl6656 Mar 2, 2025

0

1

20

1

+417

Algebra

Let
p(a, b, c) = a^5 + b^5 + c^5 + k(a^3 + b^3 + c^3)(a^4 + b^4 + c^4 - 2a^2 b^2 - 2a^2 c^2 - 2b^2 c^2).
Find the constant k so that p(a,b,c) is divisible by a + b + c.

nathanl6656 Mar 2, 2025

+1

1

15

1

+490

Algebra

If z is a complex number satisfying
z + \frac{1}{z} = \sqrt{5},
calculate
z^{10} + \frac{1}{z^{10}}. read more ..

MeIdHunter Mar 2, 2025

0

1

18

0

+490

Algebra

MeIdHunter Mar 2, 2025

0

1

17

1

+490

Algebra

In this multi-part problem, we will consider this system of simultaneous equations:
3x + 4y + 30z = -60,
2xy + 42xz - 16yz = 68,
5xyz = 56.
Let a = x/2, b = 5y and c = -4z.
Determine the monic cubic polynomial in terms of read more ..

In this multi-part problem, we will consider this system of simultaneous equations:
3x + 4y + 30z = -60,
2xy + 42xz - 16yz = 68,
5xyz = 56.
Let a = x/2, b = 5y and c = -4z.
Determine the monic cubic polynomial in terms of read more ..

MeIdHunter Mar 2, 2025

0

1

13

0

+490

Algebra

MeIdHunter Mar 2, 2025

0

1

20

0

+970

Algebra

If (x,y) satisfies the simultaneous equations
3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.
x^2 - y^2 = 7 + 4xy
where x and y may be complex numbers, determine all possible values of y^2.

gnistory Mar 2, 2025

0

1

12

0

+970

Algebra

Factor 3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.

gnistory Mar 2, 2025

0

28

0

+970

Algebra

Factor x^2 - 2x - y^2 + 2yz + 5z^2 as the product of two polynomials of degree 1.

gnistory Mar 2, 2025

0

26

0

+456

Algebra

Suppose the real numbers $a$, $b$, $x$, and $y$ satisfy the equations
ax + by = 3,
ax^2 + by^2 = 5,
ax^3 + by^3 = 17,
ax^4 + by^4 = 23.<#> Evaluate ax^5 + by^5.

Jasmo Mar 2, 2025

0

23

0

+456

Algebra

Let a, b, c, and d be distinct real numbers such that
a = \sqrt{4 + \sqrt{5 + a}},
b = \sqrt{4 - \sqrt{7 + b}},
c = \sqrt{4 + \sqrt{9 - c}},
d = \sqrt{4 - \sqrt{11 - d}}.
Compute abcd.

Jasmo Mar 2, 2025

0

25

0

+456

Algebra

If a_1, a_2, ..., a_19 satisfy
a_1 + a_2 + a_3 + a_4 = 1,
a_2 + a_3 + a_4 + a_5 = 2,
a_3 + a_4 + a_5 + a_6 = 3,
...
a_{16} + a_{17} + a_{18} + a_{19} = 16,
a_{17} + a_{18} + a_{19} + read more ..

Jasmo Mar 1, 2025

0

24

0

+456

Algebra

Let a = 3(x - y), b = 3(y - z), and c = 3(z - x), where x, y, z are real numbers, and assume ab + ac + bc \ne 0. Compute
(a^3 + b^3 + c^3)/(ab + ac + bc).

Jasmo Mar 1, 2025

0

1

13

1

+900

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

Fill in the blanks, to complete the factorizaion:
(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)

eramsby1O1O Mar 1, 2025

0

21

0

+900

Algebra

eramsby1O1O Mar 1, 2025

0

22

1

+900

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 5n - mn + 20m + 4n + ___

Let t be a root of f(x) = x^3 - x + 3. Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.

eramsby1O1O Mar 1, 2025

0

27

0

+900

Algebra

eramsby1O1O Mar 1, 2025

0

26

0

+483

Algebra

Find the foci of the ellipse shown below.
Enter your answer as a list of ordered pairs separated by commas.
Points on the ellipse are (-4,9), (-7,2), (-5,2), (-7, 1)

bIueb3rry Mar 1, 2025

0

16

0

+483

Algebra

The real numbers x and y satisfy
x^2 + y^2 - 8x + 6y + 23 = 0.
Find the largest possible value of x + y.

bIueb3rry Mar 1, 2025

0

10

0

+483

Algebra

Let f(x) be a polynomial. Find the remainder when f(x) is divided by x(x - 1)(x - 2), if f(0) = 0, f(1) = 1, and f(2) = 2.

bIueb3rry Mar 1, 2025

0

23

0

+483

Algebra

Let f(x) be a polynomial. Find the remainder when f(x) is divided by x(x - 1)(x - 2), if f(0) = 0, f(1) = 1, and f(2) = 4.

bIueb3rry Mar 1, 2025

0

30

0

+357

Algebra

Let a, b, c be the roots of p(x) = x^3 + 7x^2 + 10x - 13 - 5x^3 + 25x^2 + 44x - 15. Find the value of
1/(ab + ac) + 1/(ab + bc) + 1/(ac + bc)

cwimie Mar 1, 2025

0

18

1

+357

Algebra

Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 0, what is b?

Find all (real or nonreal) x satisfying
(x - 3)^4 + (x - 5)^4 = -8 + 6(x - 3)(x - 5)^3 - 11(x - 3)^3 (x - 5).

cwimie Mar 1, 2025

0

12

1

+357

Algebra

Express x^8 + x^4 y^4 - x^6 y^2 + x^3 y^5 - 4xy^7 - y^8 as the product of two polynomials of degree 2, and a polynomial of degree 4.

cwimie Mar 1, 2025

0

29

0

+456

Algebra

Jasmo Feb 27, 2025

0

18

1

+456

Algebra

Find the constant k so that
2x^2 + 5xy - 8y^2 + 7x + 25y + k
can be expressed as the product of two linear factors of the form ax + by + c.

Let
p(a, b, c) = a^5 + b^5 + c^5 + k(a^3 + b^3 + c^3)(a^4 + b^4 + c^4 - 2a^2 b^2 - 2a^2 c^2 - 2b^2 c^2).
Find the constant k so that p(a,b,c) is divisible by a + b + c.

Jasmo Feb 27, 2025

0

26

0

+357

Algebra

cwimie Feb 27, 2025

0

8

1

+357

Algebra

If z is a complex number satisfying
z + \frac{1}{z} = \sqrt{2},
calculate
z^{10} + \frac{1}{z^{10}}. read more ..

In this multi-part problem, we will consider this system of simultaneous equations:
3x + 4y + 30z = -60,
2xy + 42xz - 16yz = 68,
5xyz = 56.
Let a = x/2, b = 5y and c = -4z.
Determine the monic cubic polynomial in terms of read more ..

cwimie Feb 27, 2025

0

14

1

+357

Algebra

In this multi-part problem, we will consider this system of simultaneous equations:
3x + 4y + 30z = -60,
2xy + 42xz - 16yz = 68,
5xyz = 56.
Let a = x/2, b = 5y and c = -4z.
Determine the monic cubic polynomial in terms of read more ..

cwimie Feb 26, 2025

0

23

1

+357

Algebra

If (x,y) satisfies the simultaneous equations
3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.
x^2 - y^2 = 7 + 4xy
where x and y may be complex numbers, determine all possible values of y^2.

cwimie Feb 26, 2025

0

21

0

+456

Algebra

Jasmo Feb 26, 2025

0

13

0

+456

Algebra

Factor 3xy - 4x^2 + 18y - 24x + 5x^2*y - 8y^3 + 20.

Jasmo Feb 26, 2025

0

19

0

+456

Algebra

Let r_1, r_2, r_3, r_4, and r_5 be the complex roots of x^5 - 4x^2 + 7x - 1 = 0. Compute
(r_1^2 + r_1^6 + 2)(r_2^2 + r_2^6 + 2)(r_3^2 + r_3^6 + 2)(r_4^2 + r_4^6 + 2)(r_5^2 + r_5^6 + 2)

Jasmo Feb 25, 2025

0

23

0

+456

Algebra

Fill in the blanks, to complete the factorizaion:
(a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2 = (a + ___)(a + ___)(a + ___)(a + ___)

Jasmo Feb 25, 2025

0

26

0

+456

Algebra

Factor x^2 - 2x + 7 - y^2 - 10yz - 13z^2 as the product of two polynomials of degree 1.

Jasmo Feb 25, 2025

0

20

2

+456

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 7n - 3mn + 15m + 8n + ___

Factor x^2 - 2x - y^2 + 2yz + 5z^2 as the product of two polynomials of degree 1.

Jasmo Feb 25, 2025

0

21

1

+357

Algebra

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 5n - mn + 20m + 4n + ___

cwimie Feb 25, 2025

0

25

0

+357

Algebra

cwimie Feb 25, 2025

0

20

1

+357

Algebra

Let r, s, and t be solutions of the equation 3x^3 - 4x^2 - 2x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}.

history ●

cwimie Feb 25, 2025

+1

25

0

+357

Geometry

Let $WXYZ$ be a trapezoid with bases $\overline{XY}$ and $\overline{WZ}$. In this trapezoid, $\angle ZXW = 120^\circ$, $\angle XWZ = 60^\circ$, and $\angle XYZ = 120^\circ$. Find $\angle YXZ$, in degrees.

cwimie Feb 25, 2025

+1

12

1

+357

Geometry

Let $G$ be the center of equilateral triangle $XYZ.$ A dilation centered at $G$ with scale factor $-1$ is applied to triangle $XYZ,$ to obtain triangle $X'Y'Z'.$ Let $A$ be the area of the region that is contained in both triangles $XYZ$ and $X'Y'Z'.$ Find read more ..

Points $A$ and $B$ are on side $\overline{YZ}$ of rectangle $WXYZ$ such that $\overline{WA}$ and $\overline{WB}$ trisect $\angle ZWX$. If $WX = 2$ and $XY = 3$, then what is the area of rectangle $WXYZ$?

cwimie Feb 25, 2025

+1

18

2

+357

Geometry

In rectangle $WXYZ$, $A$ is on side $\overline{WX}$, $B$ is on side $\overline{YZ}$, and $C$ is on side $\overline{XY}$. If $AX = 15$, $BY = 20$, $\angle ACB= 60^\circ$, and $CY = 3 \cdot CX$, then find $AB$.

cwimie Feb 25, 2025

+1

33

0

+357

Geometry

cwimie Feb 25, 2025

+2

22

1

+38

Trig help (no random guests allowed)

Problem:
Find AC.
[asy] size(300); pair A,B,C,O; A=origin; B=6*sqrt(3)*dir(150); C=(5,0);draw(A--B--C--cycle); dot("$A$",A,SW); dot("$B$",B,NW); dot("$C$",C,SE); label("$150^\circ$",A,dir(75)); label("$9$",A--B,SW); label("$7\sqrt{3}$",C--B,NE);read more ..

The graph shows that Anna and Boris begin walking at the same time. Find Boris's speed. Edit: I know that this forum is mostly about the Aops class ciriculum.This problem comes from a different ciriculum. As a new member who lacks the knowledge of this read more ..

skibidibrain901 Feb 25, 2025

+2

19

1

+17

HELP ME!!111

Suppose r and s are the values of x that satisfy the equation
x^2 - 2mx + (m^2 - 6m + 11) = 0
for some real number m. Find the minimum real value of (r - s)^2.

Chiccen Feb 24, 2025

0

14

1

+456

Algebra

Let r, s, and t be solutions of the equation x^3 + 2x^2 - 5x + 15 = 0. Compute
\frac{1}{r - 2s - 2t} + \frac{1}{s - 2r - 2t} + \frac{1}{t - 2r - 2s}

Jasmo Feb 24, 2025

0

11

1

+456

Algebra

Let r, s, and t be solutions of the equation x^3 - 4x^2 - 7x + 12 = 0. Compute
\frac{rs}{t^2} + \frac{rt}{s^2} + \frac{st}{r^2}

Jasmo Feb 24, 2025

0

13

3

+456

Algebra

Let t be a root of f(x) = x^3 - x + 3. Evaluate t^6 - 4t^5 + 7t^4 - 3t^2 + 10t - 13.

Jasmo Feb 24, 2025

0

23

0

+456

Algebra

Jasmo Feb 24, 2025

+1

14

1

+357

Geometry

Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point.
An equilateral triangle, a regular decagon, and a regular n-gon, all with the same side length, also completely surround a read more ..

Quadrilateral ABCD is a parallelogram. Let E be a point on \overline{AB}, and let F be the intersection of lines DE and BC. The area of triangle EBF is 1, and the area of triangle BCE is 2. Find the area of parallelogram ABCD.

cwimie Feb 24, 2025

0

14

1

+357

Geometry

In trapezoid EFGH, \overline{EF} \parallel \overline{GH}, and P is the point on \overline{EH} such that EP:PH = 1:2. If the area of triangle PEF is 6, and the area of triangle PGH is 6, then find the area of trapezoid EFGH.

cwimie Feb 24, 2025

+1

24

1

+357

Geometry

Two tangents $\overline{PA}$ and $\overline{PB}$ are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$ Find the length $AB.$ read more ..

cwimie Feb 24, 2025

0

28

1

+357

Geometry

what is the answer 38x+74-92=-18

cwimie Feb 24, 2025

+3

1

13

1

+113

algebra!