All Questions ⁺⁰237750 Questions

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Math

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).

cooIcooIcooI17 Sep 4, 2025, 11:24 PM

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+898

Math

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).

cooIcooIcooI17 Sep 4, 2025, 11:24 PM

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+898

Math

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 ..

cooIcooIcooI17 Sep 4, 2025, 11:24 PM

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+898

Math

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).

cooIcooIcooI17 Sep 4, 2025, 11:24 PM

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+1844

Math

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 Sep 4, 2025, 6:42:37 PM

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+1844

Math

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)

blackpanther Sep 4, 2025, 6:42:11 PM

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+1844

Math

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

blackpanther Sep 4, 2025, 6:41:56 PM

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+230

Math

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\}.

HumanBemg Sep 4, 2025, 6:41:30 PM

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+230

Math

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}.

HumanBemg Sep 4, 2025, 6:41:17 PM

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+230

Math

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.

HumanBemg Sep 4, 2025, 6:41:04 PM

Sep 2, 2025

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+980

Math

Find the range of the function
h(x) = 5x^2 + 20x + 33/x^2

booboo44 Sep 2, 2025

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+980

Math

What is xy + yz + xz if
x^2 + xy + y^2 = 1
y^2 - yz = 2
x^2 + z^2 = 3 read more ..

booboo44 Sep 2, 2025

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+1844

Math

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\}.

blackpanther Sep 2, 2025

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+1844

Math

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}.

blackpanther Sep 2, 2025

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+1844

Math

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.

blackpanther Sep 2, 2025

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+898

Math

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.

cooIcooIcooI17 Sep 2, 2025

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+898

Math

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.

cooIcooIcooI17 Sep 2, 2025

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+898

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}

cooIcooIcooI17 Sep 2, 2025

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+898

Math

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.

cooIcooIcooI17 Sep 2, 2025

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+1006

Math

For parts (a)-(d), let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute pqr + pqs + prs + qrs.

gnistory Sep 2, 2025

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+4

Real life percentage problem

A store is offering a discount:
Get 50% off the third shirt when you buy two.
You also receive an additional 10% off the entire bill if it exceeds Rs 10,000.
Aisha purchases:
Three shirts for 1,250 rupees each<#> One bag costs read more ..

mycalsuite Sep 2, 2025

Sep 1, 2025

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+1006

Math

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

gnistory Sep 1, 2025

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+1006

Math

What is the area of the region of the complex plane defined by |z| < 3 + 2|z - 1|?

gnistory Sep 1, 2025

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+1006

Math

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

gnistory Sep 1, 2025

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