All Questions

+1808

Math

For a positive integer n, let
a_n = 1 + \sqrt{\frac{1}{n}} - \sqrt{\frac{1}{n + 1}
Compute the product a_1 a_2 a_3 \dotsm a_{99}.

blackpanther Oct 7, 2025

+1808

Math

Find the number of ways of arranging the numbers 1,2 ,3, 4, 5, 6 in a row so that the product of any two adjacent numbers is at least 5.

blackpanther Oct 7, 2025

+289

Math

A fair coin is tossed repeatedly until either heads comes up three times in a row or tails comes up four times in a row. What is the probability that the coin will be tossed more than 10 times? Express your answer as a common fraction.

CInsbstnkng Oct 7, 2025

+289

Math

In triangle ABC, cos A=sqrt(7/10) and cos B=sqrt(3/10) Find cos C

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CInsbstnkng Oct 7, 2025

Oct 6, 2025

AMC 10 I want to make AIME - PLEASE HELP

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 ..

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OOOOOOOO Oct 6, 2025

Oct 1, 2025

Stuck on a Tricky Series Problem

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 ..

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skylerclooney Oct 1, 2025

Sep 29, 2025

Sep 28, 2025

plz help im cooked

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 ..

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fof.urmum Sep 28, 2025

Aug 25, 2025

+1285

Math

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 ..

Bosco ●●

AnswerscorrectIy Aug 25, 2025

+1285

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?

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AnswerscorrectIy Aug 25, 2025

Aug 24, 2025

Aug 23, 2025

+943

Math

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 $ABC.$

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booboo44 Aug 23, 2025

+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 $BX$?

Bosco ●

booboo44 Aug 23, 2025

+1285

Math

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

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AnswerscorrectIy Aug 23, 2025

+67

Geometry Please answer with steps

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 ..

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bqimath Aug 23, 2025

+1285

Math

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

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AnswerscorrectIy Aug 23, 2025

Aug 22, 2025

-1

+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 $DE$?

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AnswerscorrectIy Aug 22, 2025

+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 $BD$?

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AnswerscorrectIy Aug 22, 2025

Aug 21, 2025

+970

Math

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

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gnistory Aug 21, 2025

+970

Math

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

gnistory Aug 21, 2025

Aug 20, 2025

+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

+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

Aug 19, 2025

+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 ..

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nerdboy Aug 19, 2025

+289

Help me help

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 $[ABCD]$?

CInsbstnkng Aug 19, 2025

+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

+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!

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bqimath Aug 19, 2025

+490

Math

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 ..

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MeIdHunter Aug 19, 2025

+294

how to solve

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 ..

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AUnVerifedTaxPayer Aug 19, 2025

Aug 18, 2025

+289

help me help me

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.

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CInsbstnkng Aug 18, 2025

+289

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 $\angle E$?

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CInsbstnkng Aug 18, 2025

Aug 17, 2025

+289

Help help

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).

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CInsbstnkng Aug 17, 2025

+289

Help me 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

Aug 16, 2025

+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

+190

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 $a$.

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HumanBemg Aug 16, 2025

+483

Help me

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 $A$?

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bIueb3rry Aug 16, 2025

+23

NEED HELP CPhill ASAP

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 ..

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Dragononweb2.0calc Aug 16, 2025

Aug 15, 2025

+417

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.

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nathanl6656 Aug 15, 2025

+970

Math

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 ..

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gnistory Aug 15, 2025

+943

I need answer

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

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booboo44 Aug 15, 2025

Aug 14, 2025

help me plz

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 ..

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askdjlkadjwl123 Aug 14, 2025

Aug 13, 2025

+289

help me now

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 $EFGH$.

CInsbstnkng Aug 13, 2025

Aug 11, 2025

+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

+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

+289

Help me!

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

CInsbstnkng Aug 11, 2025

+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

Aug 10, 2025

+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

Aug 9, 2025

+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

Aug 8, 2025

+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

+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

+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

Aug 7, 2025

+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

+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

Aug 5, 2025

+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.

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CInsbstnkng Aug 5, 2025

+289

Help me

CInsbstnkng Aug 5, 2025

Aug 3, 2025

+289

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

-1

+289

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

-1

+289

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.

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CInsbstnkng Aug 3, 2025

Aug 2, 2025

+67

Algebra

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bqimath Aug 2, 2025

-1

+289

Plz help

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

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CInsbstnkng Aug 2, 2025

-1

+289

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.

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CInsbstnkng Aug 2, 2025

-1

+289

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

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?

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CInsbstnkng Aug 2, 2025

-1

+289

Help!!! (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?

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CInsbstnkng Aug 2, 2025

+62

HELP!!! (From the real 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.

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Clnsbstnkng Aug 2, 2025

Aug 1, 2025

-1

+289

HELP!!!

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 ..

CInsbstnkng Aug 1, 2025

-1

+289

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

-1

+289

Help me

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

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CInsbstnkng Aug 1, 2025

-1

+289

HELP HELP

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

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CInsbstnkng Aug 1, 2025

+67

Counting & Probability

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 ..

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bqimath Aug 1, 2025

Jul 31, 2025

-1

+289

HELP ME

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.

CInsbstnkng Jul 31, 2025

+289

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 $WXYZ$?

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CInsbstnkng Jul 31, 2025

-1

+289

help help

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

-1

+289

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

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CInsbstnkng Jul 31, 2025

-1

+289

HELP!!!

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CInsbstnkng Jul 31, 2025

-1

+289

Help!

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

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CInsbstnkng Jul 31, 2025

Jul 30, 2025

+62

HELP!!!

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

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Clnsbstnkng Jul 30, 2025

-1

+289

help help

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

CInsbstnkng Jul 30, 2025

-1

+289

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

Jul 29, 2025

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