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.

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Found problems: 3597

2022 Kazakhstan National Olympiad, 4
Tags: algebra , polynomial , combinatorics
Let $P(x)$ be a polynomial with positive integer coefficients such that $deg(P)=699$. Prove that if $P(1) \le 2022$, then there exist some consecutive coefficients such that their sum is $22$, $55$, or $77$.

1998 Poland - First Round, 4
Tags: induction , algebra , polynomial , strong induction
Let $ x,y$ be real numbers such that the numbers $ x\plus{}y, x^2\plus{}y^2, x^3\plus{}y^3$ and $ x^4\plus{}y^4$ are integers. Prove that for all positive integers $ n$, the number $ x^n \plus{} y^n$ is an integer.

2009 Iran Team Selection Test, 8
Tags: polynomial , algebra
Find all polynomials $ P(x,y)$ such that for all reals $ x$ and $y$, \[P(x^{2},y^{2}) =P\left(\frac {(x + y)^{2}}{2},\frac {(x - y)^{2}}{2}\right).\]

2012 ELMO Shortlist, 7
Tags: polynomial , function , algebra
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

2004 IMO Shortlist, 4
Tags: algebra , polynomial , functional equation
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations \[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

2007 Brazil National Olympiad, 1
Tags: quadratic , induction , algebra , polynomial , limit , quadratic formula
Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.

2015 Vietnam National Olympiad, 1
Tags: polynomial , induction , algebra
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and ${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$ Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.

2019 Vietnam National Olympiad, Day 1
Tags: polynomial
For each real coefficient polynomial $f(x)={{a}_{0}}+{{a}_{1}}x+\cdots +{{a}_{n}}{{x}^{n}}$, let $$\Gamma (f(x))=a_{0}^{2}+a_{1}^{2}+\cdots +a_{m}^{2}.$$ Let be given polynomial $P(x)=(x+1)(x+2)\ldots (x+2020).$ Prove that there exists at least $2019$ pairwise distinct polynomials ${{Q}_{k}}(x)$ with $1\le k\le {{2}^{2019}}$ and each of it satisfies two following conditions: i) $\deg {{Q}_{k}}(x)=2020.$ ii) $\Gamma \left( {{Q}_{k}}{{(x)}^{n}} \right)=\Gamma \left( P{{(x)}^{n}} \right)$ for all positive initeger $n$.

2023 Indonesia TST, A
Tags: polynomial , integer polynomial , algebra
Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation: \[Q(a+b) = \frac{P(a) - P(b)}{a - b}\] $\forall a, b \in \mathbb{Z}^+$ and $a>b$

PEN Q Problems, 13
Tags: algebra , algorithm , polynomial
On Christmas Eve, 1983, Dean Jixon, the famous seer who had made startling predictions of the events of the preceding year that the volcanic and seismic activities of $1980$ and $1981$ were connected with mathematics. The diminishing of this geological activity depended upon the existence of an elementary proof of the irreducibility of the polynomial \[P(x)=x^{1981}+x^{1980}+12x^{2}+24x+1983.\] Is there such a proof?

2017 Harvard-MIT Mathematics Tournament, 1
Tags: algebra , polynomial
Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$. Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$.

2020 MMATHS, I12
Tags: trigonometry , algebra , polynomial
Let $p(x)$ be the monic cubic polynomial with roots $\sin^2(1^{\circ})$, $\sin^2(3^{\circ})$, and $\sin^2(9^{\circ})$. Suppose that $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$, where $0 <a,b \le 90$ and $a,b,n$ are positive integers. What is $a+b+n$? [i]Proposed by Andrew Yuan[/i]

2016 Postal Coaching, 5
Tags: algebra , polynomial
A real polynomial of odd degree has all positive coefficients. Prove that there is a (possibly trivial) permutation of the coefficients such that the resulting polynomial has exactly one real zero.

2019 Denmark MO - Mohr Contest, 2
Tags: algebra , polynomial
Two distinct numbers a and b satisfy that the two equations $x^{2019} + ax + 2b = 0$ and $x^{2019}+ bx + 2a = 0$ have a common solution. Determine all possible values of $a + b$.

1987 Traian Lălescu, 2.1
Tags: abstract algebra , ring theory , extension rings , superrings , algebra , polynomial
Any polynom, with coefficients in a given division ring, that is irreducible over it, is also irreducible over a given extension skew ring of it that's finite. Prove that the ring and its extension coincide.

1987 IMO, 3
Tags: algebra , polynomial , number theory , prime number
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.

2017 Costa Rica - Final Round, A1
Tags: polynomial , algebra
Let $P (x)$ be a polynomial of degree $2n$, such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$. Determine $P (2n + 1)$.

2016 Serbia Additional Team Selection Test, 1
Tags: polynomial , algebra
Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\ $P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\ Prove that $x^{2016}|P_{2016}(x)$.

2022 Belarusian National Olympiad, 10.7
Tags: algebra , polynomial
Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$

2012 Vietnam Team Selection Test, 1
Tags: algebra , polynomial , diophantine equation , number theory
Consider the sequence $(x_n)_{n\ge 1}$ where $x_1=1,x_2=2011$ and $x_{n+2}=4022x_{n+1}-x_n$ for all $n\in\mathbb{N}$. Prove that $\frac{x_{2012}+1}{2012}$ is a perfect square.

1974 IMO Longlists, 49
Tags: polynomial , calculus , integration , trigonometry , algebra
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$

2016 District Olympiad, 4
Tags: geometry , algebra , polynomial , number theory
Let $ a\ge 2 $ be a natural number. Show that the following relations are equivalent: $ \text{(i)} \ a $ is the hypothenuse of a right triangle whose sides are natural numbers. $ \text{(ii)}\quad $ there exists a natural number $ d $ for which the polynoms $ X^2-aX\pm d $ have integer roots.

2006 Turkey MO (2nd round), 3
Tags: geometry , trigonometry , circumcircle , function , algebra , polynomial , rational root theorem
Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.

2001 Manhattan Mathematical Olympiad, 3
Tags: modular arithmetic , algebra , polynomial
Let $x_1$ and $x_2$ be roots of the equation $x^2 - 6x + 1 = 0$. Prove that for any integer $n \ge 1$ the number $x_1^n + x_2^n$ is integer and is not divisible by $5$.

1998 Austrian-Polish Competition, 5
Tags: integer , integer polynomial , polynomial , algebra
Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).