Olympiad problems

2021 Saudi Arabia Training Tests, 35

Let $P (x)$ be a non constant integer polynomial and positive integer $n$. The sequence $a_0, a_1, ...$ is defined by $a_0 = n$ and $a_k = P (a_{k-1})$ for $k \ge 1$. Given that for each positive integer $b$, the sequence contains a $b$-th power of some positive integer greater than $1$. Prove that deg $P = 1$

2017 Azerbaijan EGMO TST, 3

Tags: polynomial , real root , algebra

The degree of the polynomial $P(x)$ is $2017.$ Prove that the number of distinct real roots of the equation $P(P(x)) = 0$ is not less than the number of distinct real roots of the equation $P(x) = 0.$

2018 Hong Kong TST, 1

Tags: algebra , polynomial

Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\sqrt[3]{2})=1+\sqrt[3]{2}$ and $P(1+\sqrt5)=2+3\sqrt5$?

2013 APMO, 4

Tags: polynomial , combinatorics , algebra

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

1988 Vietnam National Olympiad, 2

Tags: polynomial , algebra

Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$, $ a_{n\minus{}1} \equal{} \minus{}n$, $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$.

2008 Bulgarian Autumn Math Competition, Problem 9.3

Tags: natural number , number theory , algebra , polynomial , divisor

Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

2022 International Zhautykov Olympiad, 1

Tags: algebra , polynomial

Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities $$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$ Prove that the degrees of the three polynomials are all even.

1998 German National Olympiad, 4

Tags: polynomial , quadratic , algebra

Let $a$ be a positive real number. Then prove that the polynomial \[ p(x)=a^3x^3+a^2x^2+ax+a \] has integer roots if and only if $a=1$ and determine those roots.

2014 Contests, 2

Tags: polynomial , algebra , number theory

find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

1985 IMO Longlists, 92

Tags: algebra , polynomial , root , algorithm

Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.

2018 Romanian Master of Mathematics Shortlist, A1

Tags: algebra , polynomial

Let $m$ and $n$ be integers greater than $2$, and let $A$ and $B$ be non-constant polynomials with complex coefficients, at least one of which has a degree greater than $1$. Prove that if the degree of the polynomial $A^m-B^n$ is less than $\min(m,n)$, then $A^m=B^n$. [i]Proposed by Tobi Moektijono, Indonesia[/i]

2021 Iran Team Selection Test, 3

Tags: algebra , polynomial , inequalities , number theory , relatively prime

Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root) Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]

2008 IMC, 1

Tags: algebra , polynomial , modular arithmetic , ratio , number theory , relatively prime

Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.

1995 South africa National Olympiad, 2

Tags: algebra , polynomial , vieta , function , number theory

Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.

2013 Israel National Olympiad, 3

Tags: algebra , polynomial , arctangent

Let $p(x)=x^4-5773x^3-46464x^2-5773x+46$. Determine the sum of $\arctan$-s of its real roots.

1980 Miklós Schweitzer, 7

Tags: algebra , polynomial , trigonometry , real analysis

Let $ n \geq 2$ be a natural number and $ p(x)$ a real polynomial of degree at most $ n$ for which \[ \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ .\] Prove that then \[ |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right).\] [i]J. Szabados[/i]

1987 IMO Longlists, 69

Tags: number theory , polynomial , prime number , quadratic

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

1995 Korea National Olympiad, Day 2

Tags: polynomial , integer polynomial , greatest common divisor , prime , factorization , algebra

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

2018 Middle European Mathematical Olympiad, 2

Tags: algebra , polynomial , root

Let $P(x)$ be a polynomial of degree $n\geq 2$ with rational coefficients such that $P(x) $ has $ n$ pairwise different reel roots forming an arithmetic progression .Prove that among the roots of $P(x) $ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients .

2010 Irish Math Olympiad, 5

Tags: polynomial , algebra

Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.

2005 IberoAmerican Olympiad For University Students, 1

Tags: polynomial , analytic geometry , limit , induction , ratio , algebra

Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find \[\lim_{n\to\infty}f(n)^{1/n}\]

2018 Iran MO (3rd Round), 4

Tags: algebra , polynomial

Let $P(x)$ be a non-zero polynomial with real coefficient so that $P(0)=0$.Prove that for any positive real number $M$ there exist a positive integer $d$ so that for any monic polynomial $Q(x)$ with degree at least $d$ the number of integers $k$ so that $|P(Q(k))| \le M$ is at most equal to the degree of $Q$.

1991 Bulgaria National Olympiad, Problem 4

Tags: polynomial , algebra

Let $f(x)$ be a polynomial of degree $n$ with real coefficients, having $n$ (not necessarily distinct) real roots. Prove that for all real $x$, $$f(x)f''(x)\le f'(x)^2.$$

2015 IFYM, Sozopol, 7

Tags: polynomial , algebra

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

1988 IMO Longlists, 39

Tags: polynomial , function , functional equation , algebra

[b]i.)[/b] Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12}$ is divided by the polynomial $g(x)$? [b]ii.)[/b] If $k$ is a positive number and $f$ is a function such that, for every positive number $x, f(x^2 + 1 )^{\sqrt{x}} = k.$ Find the value of \[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number $y.$ [b]iii.)[/b] The function $f$ satisfies the functional equation $f(x) + f(y) = f(x+y) - x \cdot y - 1$ for every pair $x,y$ of real numbers. If $f(1) = 1,$ then find the numbers of integers $n,$ for which $f(n) = n.$