All Questions

+368

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 3 hours ago

+368

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 4 hours ago

+368

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 ..

Stanry Apr 13, 2025, 3:58:05 AM

+368

Counting

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

Stanry Apr 13, 2025, 12:25 AM

Apr 12, 2025

+368

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?

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Stanry Apr 12, 2025, 6:59:24 PM

+1517

Geometry

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?

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learnmgcat Apr 12, 2025

+1517

Geometry

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

learnmgcat Apr 12, 2025

+1517

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

+1517

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 ..

learnmgcat Apr 12, 2025

+802

Counting

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 ..

crimefightingvigiI Apr 12, 2025

+802

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

+802

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

+802

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

+802

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

+1782

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

+1782

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

+1782

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

Apr 11, 2025

+1782

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

Apr 10, 2025

+582

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.

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onyuIee Apr 10, 2025

+582

Counting

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

+582

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

Apr 9, 2025

+164

Counting

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

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HumanBemg Apr 9, 2025

+164

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.)

HumanBemg Apr 9, 2025

Apr 8, 2025

+164

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 8, 2025

+164

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

+921

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

+921

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

+921

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 ..

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booboo44 Apr 8, 2025

+921

Counting

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 ..

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booboo44 Apr 8, 2025

Apr 7, 2025

+950

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.

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gnistory Apr 7, 2025

+950

Algebra

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

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gnistory Apr 7, 2025

+448

Free Body Diagram

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 ..

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HiylinLink Apr 7, 2025

+368

Geometry

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?

Bosco ●●

Stanry Apr 7, 2025

Apr 5, 2025

-1

+164

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).

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HumanBemg Apr 5, 2025

+164

Geometry

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

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HumanBemg Apr 5, 2025

+164

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

+950

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)

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gnistory Apr 5, 2025

+950

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

+950

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

Apr 4, 2025

+950

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?

gnistory Apr 4, 2025

+950

Geometry

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

+921

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.

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booboo44 Apr 4, 2025

+290

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?

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AUnVerifedTaxPayer Apr 4, 2025

+368

i need 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 BC = 12, AC = 18 and BF = 3 \cdot AF. Find AB.

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Stanry Apr 4, 2025

+368

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 congruent/similar. read more ..

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Stanry Apr 4, 2025

+368

need help

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Stanry Apr 4, 2025

+71

Counting and Probability, Melody or CPhill 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 ..

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Starry Apr 4, 2025

+1517

Geometry

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.

learnmgcat Apr 4, 2025

+1517

Geometry

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

learnmgcat Apr 4, 2025

+426

Geometry

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

bIueb3rry Apr 4, 2025

+426

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

+426

Geometry

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

bIueb3rry Apr 4, 2025

Apr 3, 2025

+802

Algebra

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.

crimefightingvigiI Apr 3, 2025

Apr 2, 2025

+802

Algebra

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

crimefightingvigiI Apr 2, 2025

Apr 1, 2025

+1729

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

+164

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

Mar 31, 2025

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 ..

xhxmsja Mar 31, 2025

Mar 30, 2025

+44

Algebra

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

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XenosmiIus Mar 30, 2025

+44

Algebra

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

XenosmiIus Mar 30, 2025

+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)?

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XenosmiIus Mar 30, 2025

+44

Algebra

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.

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XenosmiIus Mar 30, 2025

-1

+478

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 $DE$?

●●●

MeIdHunter Mar 30, 2025

-1

+478

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 $BD$?

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MeIdHunter Mar 30, 2025

+478

Geometry

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

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MeIdHunter Mar 30, 2025

help please

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 ..

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whatthedonut Mar 30, 2025

+44

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.

XenosmiIus Mar 30, 2025

+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

+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

+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

+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

+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

+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

Mar 29, 2025

+18

Interesting Range Question; Can use polynomial long division to solve

Find the range of the function

Enter your answer in interval notation.

●●●

Xenosmilus Mar 29, 2025

+164

Algebra

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

HumanBemg Mar 29, 2025

+164

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

Mar 27, 2025

+1782

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

+164

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

+164

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

+164

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

+164

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

Mar 26, 2025

+1782

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).

●

blackpanther Mar 26, 2025

+1782

Algebra

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

Mar 25, 2025

+1782

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 ..

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blackpanther Mar 25, 2025

+1782

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

+1782

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

+1782

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

+1782

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

+164

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

+164

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

+164

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

+164

Algebra

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

HumanBemg Mar 25, 2025

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 ..

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sPpBCdIT Mar 25, 2025

Mar 24, 2025

+339

Algebra

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

jaekg Mar 24, 2025

+339

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 ..

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jaekg Mar 24, 2025

+339

Algebra

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

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jaekg Mar 24, 2025

+339

Algebra

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

jaekg Mar 24, 2025

Mar 23, 2025

+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 ..

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HumenBeing Mar 23, 2025

Mar 22, 2025

+802

Algebra

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).

crimefightingvigiI Mar 22, 2025

+802

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

Mar 21, 2025

+164

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

+164

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

+164

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

Mar 20, 2025

+116

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).

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coIinqiu Mar 20, 2025

+1208

Algebra

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).

AnswerscorrectIy Mar 20, 2025

+1208

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

+1208

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

+1208

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

+1208

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

Mar 19, 2025

+1208

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

+1208

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

+1208

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

+1208

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

+116

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

+116

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

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