All Questions ⁺⁰237357 Questions

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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

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5

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+164

Algebra

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

HumanBemg Mar 25, 2025

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5

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 ..

sPpBCdIT Mar 25, 2025

Mar 24, 2025

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+339

Algebra

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

jaekg Mar 24, 2025

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+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 ..

jaekg Mar 24, 2025

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1

+339

Algebra

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

jaekg Mar 24, 2025

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2

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+339

Algebra

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

jaekg Mar 24, 2025

Mar 23, 2025

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+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 ..

HumenBeing Mar 23, 2025

Mar 22, 2025

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3

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+806

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

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2

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+806

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

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5

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+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

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3

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+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

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4

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+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

+1

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3

+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).

coIinqiu Mar 20, 2025

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+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

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+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

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+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

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+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

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+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

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2

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+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

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2

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+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

0

2

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+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

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1

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+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

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3

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+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

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1

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+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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