Olympiad problems

2017 IFYM, Sozopol, 8

Tags: algebra , Polynomials

Find all polynomials $P\in \mathbb{R}[x]$, for which $P(P(x))=\lfloor P^2 (x)\rfloor$ is true for $\forall x\in \mathbb{Z}$.

PEN Q Problems, 6

Tags: algebra , polynomial , calculus , derivative , binomial coefficients , binomial theorem , Polynomials

Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.

PEN Q Problems, 2

Tags: algebra , polynomial , calculus , integration , pigeonhole principle , Polynomials

Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.

2013 BMT Spring, P2

Tags: Polynomials , algebra , inequalities

If $f(x)=x^n-7x^{n-1}+17x^{n-2}+a_{n-3}x^{n-3}+\ldots+a_0$ is a real-valued function of degree $n>2$ with all real roots, prove that no root has value greater than $4$ and at least one root has value less than $0$ or greater than $2$.

2013 BMT Spring, 5

Tags: algebra , Polynomials

Consider the roots of the polynomial $x^{2013}-2^{2013}=0$. Some of these roots also satisfy $x^k-2^k=0$, for some integer $k<2013$. What is the product of this subset of roots?

2009 District Olympiad, 4

Tags: superior algebra , Polynomials , finite field , probability

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

2013 USA Team Selection Test, 4

Tags: algebra , polynomial , function , abstract algebra , number theory , Polynomials

Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.

2018 APMO, 5

Tags: Polynomials , algebra , APMO , number theory

Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.

2018 Brazil Team Selection Test, 5

Tags: Polynomials , algebra , APMO , number theory

Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.

2020 USA IMO Team Selection Test, 5

Tags: number theory , TST , Polynomials , Integer Polynomial , inequalities , TST2020

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]

2004 239 Open Mathematical Olympiad, 1

Tags: function , algebra , polynomial , algebra solved , Polynomials

Given non-constant linear functions $p_1(x), p_2(x), \dots p_n(x)$. Prove that at least $n-2$ of polynomials $p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1$ have a real root.

2013 BMT Spring, 6

Tags: algebra , Polynomials , trigonometry

The [i]minimal polynomial[/i] of a complex number $r$ is the unique polynomial with rational coefficients of minimal degree with leading coefficient $1$ that has $r$ as a root. If $f$ is the minimal polynomial of $\cos\frac\pi7$, what is $f(-1)$?

1981 IMO Shortlist, 13

Tags: algebra , polynomial , Polynomials , IMO Shortlist

Let $P$ be a polynomial of degree $n$ satisfying \[P(k) = \binom{n+1}{k}^{-1} \qquad \text{ for } k = 0, 1, . . ., n.\] Determine $P(n + 1).$

2016 Israel National Olympiad, 5

Tags: algebra , number theory , Fibonacci , Fibonacci sequence , Polynomials , degree , polynomial

The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$. Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.

1964 Spain Mathematical Olympiad, 1

Tags: Polynomials , algebra

Given the equation $x^2+ax+1=0$, determine: a) The interval of possible values for $a$ where the solutions to the previous equation are not real. b) The loci of the roots of the polynomial, when $a$ is in the previous interval.

2020 ELMO Problems, P5

Tags: number theory , algebra , Polynomials

Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that [list] [*]each row contains $n$ distinct consecutive integers in some order, [*]each column contains $m$ distinct consecutive integers in some order, and [*]each entry is less than or equal to $s$. [/list] [i]Proposed by Ankan Bhattacharya.[/i]

2001 IMC, 1

Tags: Polynomials

Let $r, s \geq 1$ be integers and $a_{0}, a_{1}, . . . , a_{r-1}, b_{0}, b_{1}, . . . , b_{s-1} $ be real non-negative numbers such that $(a_0+a_1x+a_2x^2+. . .+a_{r-1}x^{r-1}+x^r)(b_0+b_1x+b_2x^2+. . .+b_{s-1}x^{s-1}+x^s) =1 +x+x^2+. . .+x^{r+s-1}+x^{r+s}$. Prove that each $a_i$ and each $b_j$ equals either $0$ or $1$.

2017 China Team Selection Test, 6

Tags: number theory , Polynomials , prime numbers , algebra , polynomial

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

2024 India IMOTC, 17

Tags: Polynomials , number theory , algebra , polynomial

Fix a positive integer $a > 1$. Consider triples $(f(x), g(x), h(x))$ of polynomials with integer coefficients, such that 1. $f$ is a monic polynomial with $\deg f \ge 1$. 2. There exists a positive integer $N$ such that $g(x)>0$ for $x \ge N$ and for all positive integers $n \ge N$, we have $f(n) \mid a^{g(n)} + h(n)$. Find all such possible triples. [i]Proposed by Mainak Ghosh and Rijul Saini[/i]

2023 Israel TST, P3

Tags: TST , number theory , Polynomials , composition , algebra

Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?

2023 Indonesia TST, A

Tags: algebra , polynomial , Polynomials , Integer Polynomial , integer polynomials , Indonesia , Indonesia TST

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 $a>b$

2021 India National Olympiad, 6

Tags: Polynomials , functions , Real Roots , algebra , INMO

Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions: [list] [*] $f$ maps the zero polynomial to itself, [*] for any non-zero polynomial $P \in \mathbb{R}[x]$, $\text{deg} \, f(P) \le 1+ \text{deg} \, P$, and [*] for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots. [/list] [i]Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha[/i]

1985 Traian Lălescu, 1.1

Tags: Divisibility , algebra , Polynomials , polynomial

Prove that for all $ n\ge 2 $ natural numbers there exist $ a_n\in\mathbb{Q} $ such that $$ X^{2n}+a_nX^n+1\Huge\vdots X^2+\frac{1}{2}X+1, $$ and that there isn´t any $ a_n\in\mathbb{R}\setminus\mathbb{Q} $ with this property.

2013 VJIMC, Problem 3

Tags: Sets , number theory , Polynomials

Let $S$ be a finite set of integers. Prove that there exists a number $c$ depending on $S$ such that for each non-constant polynomial $f$ with integer coefficients the number of integers $k$ satisfying $f(k)\in S$ does not exceed $\max(\deg f,c)$.

2018 China Team Selection Test, 1

Tags: polynomial , Polynomials , algebra , Rational Root Theorem , Sequence

Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$