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

1987 IMO Shortlist, 23
Tags: number theory , polynomial , irrational number , algebra , modular arithmetic
Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, \[[r^m] \equiv -1 \pmod k .\] [i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients. [i]Proposed by Yugoslavia.[/i]

2021 AMC 12/AHSME Fall, 23
Tags: algebra , polynomial
A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)?$ $\textbf{(A) }\dfrac5{16} \qquad \textbf{(B) }\dfrac12 \qquad \textbf{(C) }\dfrac58 \qquad \textbf{(D) }1 \qquad \textbf{(E) }\dfrac98$

1966 Poland - Second Round, 2
Tags: polynomial , algebra , cubic
Prove that if two cubic polynomials with integer coefficients have an irrational root in common, then they have another common irrational root.

2009 China Team Selection Test, 3
Tags: algebra , polynomial , binomial coefficient , induction , modular arithmetic
Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

2013 Princeton University Math Competition, 8
Tags: algebra , absolute value , polynomial
If $x,y$ are real, then the $\textit{absolute value}$ of the complex number $z=x+yi$ is \[|z|=\sqrt{x^2+y^2}.\] Find the number of polynomials $f(t)=A_0+A_1t+A_2t^2+A_3t^3+t^4$ such that $A_0,\ldots,A_3$ are integers and all roots of $f$ in the complex plane have absolute value $\leq 1$.

2021 China Team Selection Test, 4
Tags: algebra , number theory , modular arithmetic , polynomial
Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

2010 Germany Team Selection Test, 3
Tags: number theory , algebra , permutation , polynomial
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$. [i]Proposed by Jozsef Pelikan, Hungary[/i]

1999 Tuymaada Olympiad, 2
Tags: algebra , polynomial
Find all polynomials $P(x)$ such that \[ P(x^3+1)=P(x^3)+P(x^2). \] [i]Proposed by A. Golovanov[/i]

2003 India Regional Mathematical Olympiad, 6
Tags: polynomial , algebra , quadratic
Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$.

2004 VJIMC, Problem 4
Tags: real analysis , polynomial , algebra
Let $f:\mathbb R\to\mathbb R$ be an infinitely differentiable function. Assume that for every $x\in\mathbb R$ there is an $n\in\mathbb N$ (depending on $x$) such that $$f^{(n)}(x)=0.$$Prove that $f$ is a polynomial.

1983 AMC 12/AHSME, 6
Tags: algebra , polynomial
When \[x^5, \quad x+\frac{1}{x}\quad \text{and}\quad 1+\frac{2}{x} + \frac{3}{x^2}\] are multiplied, the product is a polynomial of degree $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $

2006 Spain Mathematical Olympiad, 1
Tags: algebra , integer root , integer polynomial , polynomial
Let $P(x)$ be a polynomial with integer coefficients. Prove that if there is an integer $k$ such that none of the integers $P(1),P(2), ..., P(k)$ is divisible by $k$, then $P(x)$ does not have integer roots.

1967 Spain Mathematical Olympiad, 8
Tags: algebra , analysis , derivative , polynomial
To obtain the value of a polynomial of degree $n$, whose coefficients are $$a_0, a_1, . . . ,a_n$$ (starting with the term of highest degree), when the variable $x$ is given the value $b$, the process indicated in the attached flowchart can be applied, which develops the actions required to apply Ruffini's rule. It is requested to build another flowchart analogous that allows to express the calculation of the value of the derivative of the given polynomial, also for $x = b$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/27563a0e97e74553a270fcd743f22176aed83b.png[/img]

2010 Purple Comet Problems, 25
Tags: number theory , relatively prime , polynomial , algebra
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$. There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$. Find $m+n$.

2013 Peru MO (ONEM), 1
Tags: number theory , algebra , polynomial
We define the polynomial $$P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x.$$ Find the largest prime divisor of $P (2)$.

2010 Germany Team Selection Test, 3
Tags: number theory , size , polynomial
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]

1996 Austrian-Polish Competition, 8
Tags: functional , polynomial , algebra , functional equation
Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.

2010 AMC 12/AHSME, 6
Tags: algebra , vieta , polynomial
A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

2013 Baltic Way, 19
Tags: algebra , number theory , polynomial , modular arithmetic
Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\ge 1$. Can $a_0$ be chosen so that $a_{54}$ is a multiple of $2013$?

2008 Bulgaria Team Selection Test, 3
Tags: floor function , algebra , polynomial , combinatorics , induction , ceiling function
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

2002 IMO Shortlist, 6
Tags: algebra , polynomial , laurentiu panaitopol
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer. [i]Laurentiu Panaitopol, Romania[/i]

2005 Taiwan National Olympiad, 3
Tags: algebra , function , polynomial
$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.

2010 Bosnia and Herzegovina Junior BMO TST, 2
Tags: minimization , polynomial , algebra
Let us consider every third degree polynomial $P(x)$ with coefficients as nonnegative positive integers such that $P(1)=20$. Among them determine polynomial for which is: $a)$ Minimal value of $P(4)$ $b)$ Maximal value of $P(3)/P(2)$

2000 Tuymaada Olympiad, 3
Tags: trigonometry , polynomial , algebra , modular arithmetic
Polynomial $ P(t)$ is such that for all real $ x$, \[ P(\sin x) \plus{} P(\cos x) \equal{} 1. \] What can be the degree of this polynomial?

2007 IMC, 1
Tags: polynomial , quadratic , modular arithmetic , algebra
Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.