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

2015 India PRMO, 5
Tags: integer , polynomial , algebra
$5.$ Let $P(x)$ be a non - zero polynomial with integer coefficients. If $P(n)$ is divisible by $n$ for each integer polynomial $n.$ What is the value of $P(0) ?$

2016 Latvia Baltic Way TST, 3
Tags: algebra , polynomial
Given a polynomial $P$ of degree $2016$ with real coefficients and a quadratic polynomial $Q$ with real coefficients. Is it possible that the roots of the polynomial $P (Q(x))$ are exactly all these numbers: $$-2015, -2014, . . . , -2, -1, 1, 2, . . . , 2016, 2017?$$

2022 Moldova EGMO TST, 4
Tags: algebra , polynomial , abstract algebra , modular arithmetic
Prove that there exists an integer polynomial $P(X)$ such that $P(n)+4^n \equiv 0 \pmod {27}$. for all $n \geq 0$.

2014 PUMaC Individual Finals A, 3
Tags: modular arithmetic , induction , polynomial , number theory , relatively prime , algebra
There are $n$ coins lying in a circle. Each coin has two sides, $+$ and $-$. A $flop$ means to flip every coin that has two different neighbors simultaneously, while leaving the others alone. For instance, $++-+$, after one $flop$, becomes $+---$. For $n$ coins, let us define $M$ to be a $perfect$ $number$ if for any initial arrangement of the coins, the arrangement of the coins after $m$ $flops$ is exactly the same as the initial one. (a) When $n=1024$, find a perfect number $M$. (b) Find all $n$ for which a perfect number $M$ exist.

1986 Czech And Slovak Olympiad IIIA, 2
Tags: algebra , polynomial , integer polynomial , integer
Let $P(x)$ be a polynomial with integer coefficients of degree $n \ge 3$. If $x_1,...,x_m$ ($n\ge m\ge3$) are different integers such that $P(x_1) = P(x_2) = ... = P(x_m) = 1$, prove that $P$ cannot have integer roots$.

2022 ISI Entrance Examination, 7
Tags: polynomial , limit
Let $$P(x)=1+2 x+7 x^{2}+13 x^{3}~,\qquad x \in \mathbb{R} .$$ Calculate for all $x \in \mathbb{R},$ $$\lim _{n \rightarrow \infty}\left(P\left(\frac{x}{n}\right)\right)^{n}$$

2010 National Olympiad First Round, 27
Tags: algebra , polynomial , inequalities , vector
Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

1966 IMO Shortlist, 35
Tags: algebra , polynomial , irrational number , root
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

1992 India Regional Mathematical Olympiad, 1
Tags: polynomial , inequalities , algebra
Determine the set of integers $n$ for which $n^2+19n+92$ is a square.

2023 Moldova Team Selection Test, 4
Tags: algebra , polynomial
Polynomials $(P_n(X))_{n\in\mathbb{N}}$ are defined as: $$P_0(X)=0, \quad P_1(X)=X+2,$$ $$P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X), \quad (\forall) n\geq2.$$ Show that if $ k $ divides $m$ then $P_k(X)$ divides $P_m(X).$

2013 Iran Team Selection Test, 5
Tags: modular arithmetic , algebra , polynomial , quadratic , vieta , number theory
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? [i]Proposed by Mahan Malihi[/i]

2010 Contests, 1
Tags: polynomial , limit , algebra
suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

1947 Moscow Mathematical Olympiad, 128
Tags: algebra , polynomial , coefficient
Find the coefficient of $x^2$ after expansion and collecting the terms of the following expression (there are $k$ pairs of parentheses): $$((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2$$

2019 Ecuador NMO (OMEC), 1
Tags: algebra , polynomial
Find how many integer values $3\le n \le 99$ satisfy that the polynomial $x^2 + x + 1$ divides $x^{2^n} + x + 1$.

2007 ISI B.Math Entrance Exam, 3
Tags: algebra , polynomial , limit , complex number
For a natural number $n>1$ , consider the $n-1$ points on the unit circle $e^{\frac{2\pi ik}{n}}\ (k=1,2,...,n-1) $ . Show that the product of the distances of these points from $1$ is $n$.

2013 All-Russian Olympiad, 1
Tags: polynomial , algebra
Let $P(x)$ and $Q(x)$ be (monic) polynomials with real coefficients (the first coefficient being equal to $1$), and $\deg P(x)=\deg Q(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions, then $ P(x+1)=Q(x-1) $ has a real solution.

2012 IFYM, Sozopol, 3
Tags: polynomial , algebra
The polynomial $p(x)$ is of degree $9$ and $p(x)-1$ is exactly divisible by $(x-1)^{5}$. Given that $p(x) + 1$ is exactly divisible by $(x+1)^{5}$, find $p(x)$.

2015 Irish Math Olympiad, 9
Tags: algebra , polynomial , integer root , integer polynomial
Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.

2004 Harvard-MIT Mathematics Tournament, 10
Tags: algebra , polynomial , difference of squares , special factorizations
There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.

2000 Slovenia National Olympiad, Problem 2
Tags: algebra , polynomial , inequalities
Consider the polynomial $p(x)=a_nx^n+\ldots+a_1x+a_0$ with real coefficients such that $0\le a_i\le a_0$ for each $i=1,2,\ldots,n$. If $a$ is the coefficient of $x^{n+1}$ in the polynomial $q(x)=p(x)^2$, prove that $2a\le p(1)^2$.

PEN F Problems, 15
Tags: rational number , trigonometry , algebra , polynomial
Find all rational numbers $k$ such that $0 \le k \le \frac{1}{2}$ and $\cos k \pi$ is rational.

2007 Finnish National High School Mathematics Competition, 2
Tags: polynomial , algebra
Determine the number of real roots of the equation \[x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.\]

2017 Harvard-MIT Mathematics Tournament, 6
Tags: algebra , polynomial , floor function
A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$.

Russian TST 2020, P3
Tags: algebra , polynomial
A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$ Prove that there exists a polynomial $F(t)$ in one variable such that $$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$

2023 Iberoamerican, 6
Tags: algebra , polynomial
Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-[i]representable[/i] if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-[i]representable[/i], then all the even integers are $P$-[i]representable[/i] or all the odd integers are $P$-[i]representable[/i].