Olympiad problems

2009 Finnish National High School Mathematics Competition, 2

A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?

2018 Estonia Team Selection Test, 12

Tags: polynomial , divisible

We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$. Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.

1996 IMO Shortlist, 6

Tags: polynomial , algebra , function

Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that \[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\] for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$ Note: This is variant A6' of the three variants given for this problem.

2020 Iranian Our MO, 6

Tags: algebra , functional equation , polynomial

Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$ [i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]

2010 Contests, 1

Tags: limit , algebra , polynomial

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)

2023 Quang Nam Province Math Contest (Grade 11), Problem 3

Tags: polynomial , algebra

Given a polynomial $P(x)$ with real coefficents satisfying:$$P(x).P(x+1)=P(x^2+x+1),\forall x\in \mathbb{R}.$$ Prove that: $\deg(P)$ is an even number and find $P(x).$

2010 Contests, 3

Tags: polynomial , algebra

Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

2011 Czech and Slovak Olympiad III A, 4

Tags: quadratic , algebra , polynomial

Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$, $b\ge 2$, $c\ge 2$. Adam and Boris play the following game. They alternately take turns with Adam first. On Adam’s turn, he can choose one of the polynomial’s coefficients and replace it with the sum of the other two coefficients. On Boris’s turn, he can choose one of the polynomial’s coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$, $b$ and $c$, determine who has a winning strategy.

2009 Kazakhstan National Olympiad, 6

Tags: abstract algebra , polynomial , algebra

Let $P(x)$ be polynomial with integer coefficients. Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$

2012 Online Math Open Problems, 30

Tags: polynomial , number theory , euler , algebra , induction , modular arithmetic

Let $P(x)$ denote the polynomial \[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$. [i]Victor Wang.[/i]

2021 Middle European Mathematical Olympiad, 2

Tags: algebra , polynomial , floor function

Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor)=\lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$ [list=a] [*] there exists an n-pretty polynomial; [*] any $n$-pretty polynomial has a degree of at least $\tfrac{2n+1}{3}$. [/list] ([i]Remark.[/i] For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)

1997 Austrian-Polish Competition, 5

Tags: polynomial , prime , integer polynomial , cubic

Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.

2013 Canada National Olympiad, 1

Tags: polynomial , vieta , algebra

Determine all polynomials $P(x)$ with real coefficients such that \[(x+1)P(x-1)-(x-1)P(x)\] is a constant polynomial.

2000 Iran MO (3rd Round), 3

Tags: algebra , polynomial

Prove that for every natural number $ n$ there exists a polynomial $ p(x)$ with integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .

2006 Hanoi Open Mathematics Competitions, 8

Tags: algebra , polynomial

Find all polynomials P(x) such that P(x)+P(1/x)=x+1/x

2024 Thailand October Camp, 6

Tags: algebra , polynomial

A polynomial $A(x)$ is said to be [i]simple[/i] if $A(x)$ is divisible by $x$ but not divisible by $x^2$. Suppose that a polynomial $P(x)$ has a simple polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists a simple polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{2023}$.

1984 IMO Shortlist, 12

Tags: polynomial , algebra , divisibility , number theory , modular arithmetic

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 $7^7$.

2012 All-Russian Olympiad, 4

Tags: algebra , polynomial , combinatorics

Initially there are $n+1$ monomials on the blackboard: $1,x,x^2, \ldots, x^n $. Every minute each of $k$ boys simultaneously write on the blackboard the sum of some two polynomials that were written before. After $m$ minutes among others there are the polynomials $S_1=1+x,S_2=1+x+x^2,S_3=1+x+x^2+x^3,\ldots ,S_n=1+x+x^2+ \ldots +x^n$ on the blackboard. Prove that $ m\geq \frac{2n}{k+1} $.

1998 Miklós Schweitzer, 2

Tags: real analysis , polynomial

For any polynomial f, denote by $P_f$ the number of integers n for which f(n) is a (positive) prime number. Let $q_d = max P_f$ , where f runs over all polynomials with integer coefficients with degree d and reducible over $\mathbb{Q}$. Prove that $\forall d\geq 2$ , $q_d = d$.

2006 Germany Team Selection Test, 1

Tags: composite number , prime number , algebra , polynomial , number theory

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

1984 IMO Longlists, 49

Tags: algebra , number theory , polynomial

Let $n > 1$ and $x_i \in \mathbb{R}$ for $i = 1,\cdots, n$. Set \[S_k = x_1^k+ x^k_2+\cdots+ x^k_n\] for $k \ge 1$. If $S_1 = S_2 =\cdots= S_{n+1}$, show that $x_i \in \{0, 1\}$ for every $i = 1, 2,\cdots, n.$

2004 Putnam, B1

Tags: algebra , rational root theorem , number theory , relatively prime , polynomial

Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are all integers.

2013 Baltic Way, 2

Tags: polynomial , algebra

Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies \[P(x_1)=P(x_2)= \cdots = P(x_k)=54\] \[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\] Determine the maximal value of $kn$.

2012 Iran MO (3rd Round), 4

Tags: number theory , ring theory , modular arithmetic , polynomial , algebra

$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. [b]a)[/b] Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. [b]b)[/b] If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. [i]Proposed by Mohammad Gharakhani[/i]

2018 China Team Selection Test, 5

Tags: algebra , polynomial

Suppose the real number $\lambda \in \left( 0,1\right),$ and let $n$ be a positive integer. Prove that the modulus of all the roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ are $1.$