This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 9

2019 Vietnam National Olympiad, Day 2
Tags: factoring polynomials , polynomial , algebra
Consider polynomial $f(x)={{x}^{2}}-\alpha x+1$ with $\alpha \in \mathbb{R}.$ a) For $\alpha =\frac{\sqrt{15}}{2}$, let write $f(x)$ as the quotient of two polynomials with nonnegative coefficients. b) Find all value of $\alpha $ such that $f(x)$ can be written as the quotient of two polynomials with nonnegative coefficients.

2015 AIME Problems, 3
Tags: prime number , factoring polynomials , number theory
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

2022 Bulgarian Spring Math Competition, Problem 8.1
Tags: algebra , factoring polynomials , polynomial
Let $P=(x^4-40x^2+144)(x^3-16x)$. $a)$ Factor $P$ as a product of irreducible polynomials. $b)$ We write down the values of $P(10)$ and $P(91)$. What is the greatest common divisor of the two numbers?

1993 IMO, 1
Tags: irreducible , factoring polynomials , irreducible polynomial , algebra , polynomial
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$

1993 IMO Shortlist, 7
Tags: algebra , irreducible , factoring polynomials , irreducible polynomial , polynomial
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$

2016 CCA Math Bonanza, I9
Tags: algebra , factoring polynomials , polynomial
Let $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$. Determine the number of monic polynomials $Q\left(x\right)$ with integer coefficients such that $\frac{P\left(X\right)}{Q\left(X\right)}$ is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient $1$ (so $x^3-4x+5$ is one but not $5x^3-4x^2+1$ or $x^2+3x^3$). [i]2016 CCA Math Bonanza Individual #9[/i]

2013 Stanford Mathematics Tournament, 9
Tags: algebra , factoring polynomials , polynomial
Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}, b=\sqrt{3}-\sqrt{5}+\sqrt{7}, c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate \[\frac{a^4}{(a-b)(a-c)}+\frac{b^4}{(b-c)(b-a)}+\frac{c^4}{(c-a)(c-b)}.\]

2013 Balkan MO Shortlist, A3
Tags: polynomial , algebra , factoring polynomials , integer polynomial
Prove that the polynomial $P (x) = (x^2- 8x + 25) (x^2 - 16x + 100) ... (x^2 - 8nx + 25n^2)- 1$, $n \in N^*$, cannot be written as the product of two polynomials with integer coefficients of degree greater or equal to $1$.

2020 OMpD, 1
Tags: algebra , factoring polynomials
Let $a, b, c$ be real numbers such that $a + b + c = 0$. Given that $a^3 + b^3 + c^3 \neq 0$, $a^2 + b^2 + c^2 \neq 0$, determine all possible values for: $$\frac{a^5 + b^5 + c^5}{(a^3 + b^3 + c^3)(a^2 + b^2 + c^2)}$$