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

2013 AIME Problems, 12
Tags: algebra , polynomial , trigonometry , vieta , complex number
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.

2008 Moldova Team Selection Test, 4
Tags: polynomial , factorial , calculus , integration , algebra
A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.

2017 Purple Comet Problems, 12
Tags: polynomial , algebra
Let $P$ be a polynomial satisfying $P(x + 1) + P(x - 1) = x^3$ for all real numbers $x$. Find the value of $P(12)$.

2016 IMO Shortlist, N3
Tags: number theory , polynomial , algebra , euclidean algorithm
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?

1995 China Team Selection Test, 3
Tags: quadratic , polynomial , algebra
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

2008 VJIMC, Problem 1
Tags: complex number , algebra , polynomial
Find all complex roots (with multiplicities) of the polynomial $$p(x)=\sum_{n=1}^{2008}(1004-|1004-n|)x^n.$$

2025 China Team Selection Test, 12
Tags: algebra , polynomial
Let \( P(x), Q(x) \) be non-constant real polynomials, such that for all positive integer \( m \), there exists a positive integer \( n \) satisfy \( P(m) = Q(n) \). Prove that (1) If \(\deg Q \mid \deg P\), then there exists real polynomial \( h(x) \) \( x \), satisfy \( P(x) = Q(h(x)) \) holds for all real number $x.$ (2) \(\deg Q \mid \deg P\).

1981 IMO Shortlist, 1
Tags: number theory , least common multiple , algebra , polynomial
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? [b]b.)[/b] For which $n>2$ is there exactly one set having this property?

1984 IMO Longlists, 40
Tags: algebra , polynomial , modular arithmetic , number theory , divisibility
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

2010 Indonesia TST, 1
Tags: geometry , ratio , trigonometry , algebra , polynomial , induction , function
Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

2002 Putnam, 6
Tags: linear algebra , matrix , algebra , polynomial , modular arithmetic , binomial theorem
Let $p$ be a prime number. Prove that the determinant of the matrix \[ \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} \] is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a$, $b$, and $c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)

1958 AMC 12/AHSME, 40
Tags: invariant , algebra , polynomial , linear algebra , matrix , vector , geometry
Given $ a_0 \equal{} 1$, $ a_1 \equal{} 3$, and the general relation $ a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n$ for $ n \ge 1$. Then $ a_3$ equals: $ \textbf{(A)}\ \frac{13}{27}\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \minus{}17$

1983 IMO Longlists, 38
Tags: polynomial , algebra
Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula \[u_{n+2 }= u_n - u_{n+1}.\] [b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined. [b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.

2003 Austrian-Polish Competition, 1
Tags: polynomial , functional , algebra , functional equation
Find all real polynomials $p(x) $ such that $p(x-1)p(x+1)= p(x^2-1)$.

2010 Brazil Team Selection Test, 2
Tags: polynomial , number theory , size
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

1980 IMO Shortlist, 5
Tags: analytic geometry , geometry , polynomial , algebra
In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

2012 AMC 12/AHSME, 23
Tags: algebra , polynomial , inequalities , triangle inequality
Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a, b, c$ and $d$ are integers, $0 \le d \le c \le b \le a \le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1$. What is the sum of all values $P(1)$ over all the polynomials with these properties? $ \textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 120 $

2025 Thailand Mathematical Olympiad, 10
Tags: algebra , polynomial , number theory
Let $n$ be a positive integer. Show that there exist a polynomial $P(x)$ with integer coefficient that satisfy the following [list] [*]Degree of $P(x)$ is at most $2^n - n -1$ [*]$|P(k)| = (k-1)!(2^n-k)!$ for each $k \in \{1,2,3,\dots,2^n\}$ [/list]

2023 Stars of Mathematics, 4
Tags: algebra , polynomial
Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\]

1988 IMO Shortlist, 2
Tags: algebra , polynomial , function , generating functions , binomial coefficient
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

1968 Yugoslav Team Selection Test, Problem 5
Tags: summation , polynomial , algebra
Let $n$ be an integer greater than $1$. Let $x\in\mathbb R$. (a) Evaluate $S(x,n)=\sum(x+p)(x+q)$, where the summation is over all pairs $(p,q)$ of different numbers from $\{1,2,\ldots,n\}$. (b) Do there exist integers $x,n$ for which $S(x,n)=0$?

2010 India Regional Mathematical Olympiad, 2
Tags: quadratic , algebra , polynomial
Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

1985 IMO Shortlist, 11
Tags: algebra , polynomial , root , algorithm
Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

2012 IMO Shortlist, A4
Tags: algebra , polynomial , number theory , rational roots , real analysis
Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, A1
Tags: algebra , polynomial , galois theory , complex number , superior algebra
Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$