coIinqiu

28 Questions
1 Answers

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+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

0

2

0

+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

0

1

0

+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

0

3

0

+116

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

2

0

+116

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

1

1

+116

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)

coIinqiu Mar 14, 2025

0

3

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+116

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}

coIinqiu Mar 14, 2025

0

3

0

+116

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

2

0

+116

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

3

0

+116

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

3

2

+116

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)

coIinqiu Mar 13, 2025

0

5

1

+116

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.

coIinqiu Mar 13, 2025

0

2

0

+116

Algebra

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

coIinqiu Mar 13, 2025

0

4

0

+116

Algebra

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

coIinqiu Mar 13, 2025

0

4

0

+116

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

2

1

+116

Algebra

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

coIinqiu Mar 12, 2025

0

1

1

+116

Algebra

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

coIinqiu Mar 12, 2025

0

2

1

+116

Algebra

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

coIinqiu Mar 12, 2025