Olympiad problems

2019 USA TSTST, 6

Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant? (A [i]Fibonacci number[/i] is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$.) [i]Nikolai Beluhov[/i]

2016 Flanders Math Olympiad, 4

Tags: algebra , polynomial , integer polynomial

Prove that there exists a unique polynomial function f with positive integer coefficients such that $f(1) = 6$ and $f(2) = 2016$.

1998 Israel National Olympiad, 6

Tags: polynomial , integer polynomial

Find all pairs $(m,n)$ of integers with $m > n > 7$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(7) = 77, p(m) = 0$, and $p(n) = 85$.

2006 Estonia Team Selection Test, 1

Tags: integer , algebra , quadratic , trinomial , integer polynomial

Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

2017 Saudi Arabia BMO TST, 2

Tags: algebra , integer polynomial , polynomial

Polynomial P(x) with integer coefficient is called [i]cube-presented[/i] if it can be represented as sum of several cube of polynomials with integer coefficients. Examples: $3x + 3x^2$ is cube-represented because $3x + 3x^2 = (x + 1)^3 +(-x)^3 + (-1)^3$. a) Is $3x^2$ a cube-represented polynomial? b). How many quadratic polynomial P(x) with integer coefficients belong to the set $\{1,2, 3, ...,2017\}$ which is cube-represented?

2019 ISI Entrance Examination, 7

Tags: polynomial , integer polynomial , algebra

Let $f$ be a polynomial with integer coefficients. Define $$a_1 = f(0)~,~a_2 = f(a_1) = f(f(0))~,$$ and $~a_n = f(a_{n-1})$ for $n \geqslant 3$. If there exists a natural number $k \geqslant 3$ such that $a_k = 0$, then prove that either $a_1=0$ or $a_2=0$.

2023 Indonesia TST, A

Tags: polynomial , algebra , integer polynomial

Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation: \[Q(a+b) = \frac{P(a) - P(b)}{a - b}\] $\forall a, b \in \mathbb{Z}^+$ and $a>b$

2020 USA IMO Team Selection Test, 5

Tags: polynomial , integer polynomial , inequalities , number theory

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]

1970 Swedish Mathematical Competition, 3

Tags: algebra , polynomial , integer polynomial

A polynomial with integer coefficients takes the value $5$ at five distinct integers. Show that it does not take the value $9$ at any integer.

2020 USA EGMO Team Selection Test, 6

Tags: algebra , polynomial , integer polynomial , periodic sequence , modular arithmetic

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

2004 Iran MO (3rd Round), 13

Tags: integer polynomial , integer sequence , algebra

Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$: \[ a_i | a_{i+1}\]

1987 Tournament Of Towns, (144) 1

Tags: polynomial , algebra , integer polynomial

Suppose $p(x)$ is a polynomial with integer coefficients. It is known that $p(a) - p(b) = 1$ (where $a$ and $b$ are integers). Prove that $a$ and $b$ differ by $1$ . (Folklore)

2011 Dutch IMO TST, 4

Tags: polynomial , integer polynomial , algebra

Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.

2017 Balkan MO Shortlist, A5

Tags: polynomial , algebra , integer polynomial , perfect power

Consider integers $m\ge 2$ and $n\ge 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .

2016 Thailand TSTST, 1

Tags: algebra , polynomial , number theory , integer polynomial

Find all polynomials $P\in\mathbb{Z}[x]$ such that $$|P(x)-x|\leq x^2+1$$ for all real numbers $x$.

1997 Austrian-Polish Competition, 5

Tags: integer polynomial , polynomial , cubic , prime

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 $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.

2014 IMO Shortlist, N6

Tags: number theory , divisibility , modulo arithmetic , counting in two ways , integer polynomial

Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$. [i]Proposed by Serbia[/i]

2014 Saudi Arabia GMO TST, 2

Tags: integer polynomial , polynomial , algebra

Let $S = \{f(a, b) | a, b = 1,2,3, 4$ and $a \ne b\}$, and consider all nonzero polynomials $p(X,Y )$ with integer coefficients such that $p(a, b) = 0$ for every element $(a,b)$ in $S$. (a) What is the minimal degree of such polynomial $p(X, Y )$ ? (b) Determine all such polynomials $p(X, Y )$ with minimal degree.

2008 Postal Coaching, 2

Tags: polynomial , integer polynomial , perfect square , algebra

Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.

1939 Moscow Mathematical Olympiad, 049

Tags: algebra , polynomial , integer polynomial , even , odd

Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.

2021 239 Open Mathematical Olympiad, 1

Tags: integer , polynomial , integer polynomial , algebra , prime

You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?

2018 Iran Team Selection Test, 3

Tags: polynomial , integer polynomial , number theory

$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions? a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $ b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $ [i]Proposed by Mojtaba Zare[/i]