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

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Found problems: 3597

2017 QEDMO 15th, 5
Tags: algebra , polynomial
For which natural numbers $n$ can the polynomial $f (x) = x^n + x^{n-1} +...+ x + 1$ as write $f (x) = g (h (x))$, where $g$ and $h$ should be real polynomials of degrees greater than $1$?

2023 IRN-SGP-TWN Friendly Math Competition, 6
Tags: functional equation , polynomial , polynomial with integer coeffi , algebra , number theory
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

2014 AMC 12/AHSME, 16
Tags: algebra , polynomial , system of equations
Let $P$ be a cubic polynomial with $P(0) = k, P(1) = 2k,$ and $P(-1) = 3k$. What is $P(2) + P(-2)$? $ \textbf{(A) }0 \qquad\textbf{(B) }k \qquad\textbf{(C) }6k \qquad\textbf{(D) }7k\qquad\textbf{(E) }14k\qquad $

2001 China Team Selection Test, 1
Tags: algebra , polynomial
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer

2015 India Regional MathematicaI Olympiad, 2
Tags: integer , polynomial , quadratic trinomial , algebra
Let \(P(x)=x^{2}+ax+b\) be a quadratic polynomial where \(a\) is real and \(b \neq 2\), is rational. Suppose \(P(0)^{2},P(1)^{2},P(2)^{2}\) are integers, prove that \(a\) and \(b\) are integers.

1984 AIME Problems, 15
Tags: algebra , polynomial , limit , algorithm
Determine $w^2+x^2+y^2+z^2$ if \[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array} \]

2025 Canada Junior National Olympiad, 5
Tags: algebra , polynomial
A polynomial $c_dx^d+c_{d-1}x^{d-1}+\dots+c_1x+c_0$ with degree $d$ is [i]reflexive[/i] if there is an integer $n\ge d$ such that $c_i=c_{n-i}$ for every $0\le i\le n$, where $c_i=0$ for $i>d$. Let $\ell\ge 2$ be an integer and $p(x)$ be a polynomial with integer coefficients. Prove that there exist reflexive polynomials $q(x)$, $r(x)$ with integer coefficients such that \[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]

1987 IMO Shortlist, 9
Tags: algebra , polynomial , analytic geometry , intersection , 3d geometry
Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ? [i]Proposed by Hungary.[/i] [hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]

2012-2013 SDML (High School), 8
Tags: algebra , polynomial
A polynomial $P$ with degree exactly $3$ satisfies $P\left(0\right)=1$, $P\left(1\right)=3$, and $P\left(3\right)=10$. Which of these cannot be the value of $P\left(2\right)$? $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2001 India IMO Training Camp, 2
Tags: function , polynomial , algebra
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

2012 Belarus Team Selection Test, 1
Tags: algebra , sum , min , irrational , polynomial , cubic , cubic equation
A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$ Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible. (E. Barabanov)

III Soros Olympiad 1996 - 97 (Russia), 10.2
Tags: number theory , algebra , polynomial
It is known that the equation $x^3 + px^2 + q = 0$ where $q$ is non-zero, has three different integer roots, the absolute values of two of which are prime numbers. Find the roots of this equation.

1988 IMO Longlists, 42
Tags: algebra , polynomial , vieta , inequality , root
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

PEN Q Problems, 6
Tags: binomial coefficient , polynomial , calculus , derivative , binomial theorem , algebra
Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.

2014 Contests, 1
Tags: algebra , polynomial , induction
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

1947 Putnam, B4
Tags: complex number , polynomial
Given $P(z)= z^2 +az +b,$ where $a,b \in \mathbb{C}.$ Suppose that $|P(z)|=1$ for every complex number $z$ with $|z|=1.$ Prove that $a=b=0.$

MathLinks Contest 4th, 1.1
Tags: algebra , polynomial
Let $a \ge 2$ be an integer. Find all polynomials $f$ with real coefficients such that $$A = \{a^{n^2} | n \ge 1, n \in Z\} \subset \{f(n) | n \ge 1, n \in Z\} = B.$$

2015 Baltic Way, 18
Tags: algebra , polynomial , number theory
Let $f(x)=x^n + a_{n-1}x^{n-1} + ...+ a_0 $ be a polynomial of degree $ n\ge 1 $ with $ n$ (not necessarily distinct) integer roots. Assume that there exist distinct primes $p_0,p_1,..,p_{n-1}$ such that $a_i > 1$ is a power of $p_i$, for all $ i=0,1,..,n-1$. Find all possible values of $ n$.

1967 Polish MO Finals, 4
Tags: algebra , polynomial
Prove that the polynomial $ x^3 + x + 1 $ is a factor of the polynomial $ P_n(x) = x^{n + 2} + (x+1)^{2n+1} $ for every integer $ n \geq 0 $.

2010 Stanford Mathematics Tournament, 10
Tags: algebra , polynomial , vieta
Find the sum of all solutions of the equation $\frac{1}{x^2-1}+\frac{2}{x^2-2}+\frac{3}{x^2-3}+\frac{4}{x^2-4}=2010x-4$

2003 IMO, 6
Tags: number theory , polynomial , divisibility
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

2016 Saint Petersburg Mathematical Olympiad, 7
Tags: algebra , polynomial , integer polynomial , integer
A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all such pairs of $(P(x),a)$.

2001 India IMO Training Camp, 1
Tags: algebra , polynomial , number theory
For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

1996 All-Russian Olympiad Regional Round, 11.4
Tags: algebra , polynomial
A polynomial $P(x)$ of degree $n$ has $n$ different real roots. What is the largest number of its coefficients that can be zero?

PEN F Problems, 2
Tags: rational number , trigonometry , polynomial , algebra
Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.