All Questions ⁺⁰237784 Questions

Sep 10, 2025

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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).

learnmgcat Sep 10, 2025, 3:50:18 PM

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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 ..

learnmgcat Sep 10, 2025, 3:50:00 PM

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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).

learnmgcat Sep 10, 2025, 3:49:40 PM

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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).

learnmgcat Sep 10, 2025, 3:49:26 PM

Sep 8, 2025

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+1022

Math

The polynomial
g(x) = x^3 - 14x^2 - 67x - 90 + 5x^3 - 20x^2 + 7x - 135
has one integer root. What is it?

gnistory Sep 8, 2025

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Math

If f is a polynomial of degree 4 such that
f(0) = 1, f(1) = 2, f(2) = -7, f(3) = 0, f(4) = 3,
then determine f(5).

gnistory Sep 8, 2025

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+1022

Math

Let f(x) be a polynomial with integer coefficients. There exist distinct integers p, q, r, s, t such that
f(p) = f(q) = f(r) = f(s) = 18
and f(t) > 18. What is the smallest possible value of f(t)?

gnistory Sep 8, 2025

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+1022

Math

Find all positive integers n for which n^2 - 19n + 59 + n^2 + 4n - 31 is a perfect square.

gnistory Sep 8, 2025

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+519

Math

Let f(x) = x^3 + ax^2 + bx + c be a polynomial. All the roots are negative integers. If a + b + c = 10, then find c.

bIueb3rry Sep 8, 2025

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Math

Let f(x) = x^3 + ax^2 + bx + c be a polynomial. All the roots of f(x) are negative integers. If a + b + c = 10, then find the roots of f(x).

bIueb3rry Sep 8, 2025

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Math

Let p(x) be a polynomial with integer coefficients. Suppose p(4) = 5 and p(-2) = 3. If p(x) has an integer root r, then find all possible values of r.

bIueb3rry Sep 8, 2025

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+519

Math

Let p(x) be a cubic polynomial. If p(0) = 0, p(1) = 1, p(2) = 2, and p(3) = 3, then compute p(4).

bIueb3rry Sep 8, 2025

Sep 7, 2025

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+1013

Math

Find all roots of the polynomial
f(x) = x^4 - 5x^3 + 5x^2 + 17x - 42 + 4x^4 + 10x^3 - 18x^2 + 2x - 5.

magenta Sep 7, 2025

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