Olympiad problems

2018 China Team Selection Test, 1

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 $f_n\left ( x \right ).$

2024 Saint Petersburg Mathematical Olympiad, 6

Tags: number theory , Polynomials , algebra , polynomial

Polynomial $P(x)$ with integer coefficients is given. For some positive integer $n$ numbers $P(0),P(1),\dots,P(2^n+1)$ are all divisible by $2^{2^n}$. Prove that values of $P(x)$ in all integer points are divisible by $2^{2^n}$.

2015 Taiwan TST Round 3, 2

Tags: IMO Shortlist , algebra , Functional inequality , Polynomials

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

1963 Bulgaria National Olympiad, Problem 2

Tags: algebra , Polynomials

It is given the equation $x^2+px+1=0$, with roots $x_1$ and $x_2$; (a) find a second-degree equation with roots $y_1,y_2$ satisfying the conditions $y_1=x_1(1-x_1)$, $y_2=x_2(1-x_2)$; (b) find all possible values of the real parameter $p$ such that the roots of the new equation lies between $-2$ and $1$.

2024 Tuymaada Olympiad, 7

Tags: algebra , Polynomials

Given are quadratic trinomials $f$ and $g$ with integral coefficients. For each positive integer $n$ there is an integer $k$ such that \[\frac{f(k)}{g(k)}=\frac{n + 1}{n}. \] Prove that $f$ and $g$ have a common root. [i] Proposed by A. Golovanov [/i]

2001 IMC, 4

Tags: roots of unity , Polynomials

$p(x)$ is a polynomial of degree $n$ with every coefficient $0 $ or $\pm1$, and $p(x)$ is divisible by $ (x - 1)^k$ for some integer $ k > 0$. $q$ is a prime such that $\frac{q}{\ln q}< \frac{k}{\ln n+1}$. Show that the complex $q$-th roots of unity must be roots of $ p(x). $

PEN Q Problems, 9

Tags: function , algebra , polynomial , calculus , integration , rational function , Polynomials

For non-negative integers $n$ and $k$, let $P_{n, k}(x)$ denote the rational function \[\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.\] Show that $P_{n, k}(x)$ is actually a polynomial for all $n, k \in \mathbb{N}$.

2001 Mongolian Mathematical Olympiad, Problem 1

Tags: functional equation , Polynomials

Prove that for every positive integer $n$ there exists a polynomial $p(x)$ of degree $n$ with real coefficients, having $n$ distinct real roots and satisfying $$p(x)p(4-x)=p(x(4-x))$$

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 $p^t\mid N$.

2021 Canadian Mathematical Olympiad Qualification, 1

Tags: Canada , Functional Equations , Polynomials , algebra , polynomial

Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.

1991 Bulgaria National Olympiad, Problem 4

Tags: algebra , Polynomials

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

2023 Romanian Master of Mathematics Shortlist, A2

Tags: interpolation , algebra , Polynomials , RMM Shortlist

Fix an integer $n \geq 2$ and let $a_1, \ldots, a_n$ be integers, where $a_1 = 1$. Let $$ f(x) = \sum_{m=1}^n a_mm^x. $$ Suppose that $f(x) = 0$ for some $K$ consecutive positive integer values of $x$. In terms of $n$, determine the maximum possible value of $K$.

2000 Romania National Olympiad, 4

Tags: Polynomials , algebra

Let $ f $ be a polynom of degree $ 3 $ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $ a,b $ and two roots $ x,y $ of $ f $ such that $ ax+by $ is rational, then all roots of $ f $ are rational.

2015 BMT Spring, P2

Tags: Polynomials , algebra

Let $f(x)$ be a nonconstant monic polynomial of degree $n$ with rational coefficents that is irreducible, meaning it cannot be factored into two nonconstant rational polynomials. Find and prove a formula for the number of monic complex polynomials that divide $f$.

1989 Bulgaria National Olympiad, Problem 3

Tags: Polynomials , algebra , irreducibility

Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that: (a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.

2007 All-Russian Olympiad Regional Round, 11.2

Tags: quadratics , algebra , polynomial , algebra proposed , Polynomials

Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|\plus{}|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}\equal{} g^{2}$ for some $ g\in\mathbb{R}[x]$.

2023 India National Olympiad, 2

Tags: algebra , Polynomials , inequalities , INMO , INMO 2023

Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$: $$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real. [i]Proposed by Prithwijit De[/i]

2021 Baltic Way, 19

Tags: number theory , number theory proposed , Polynomials

Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.

2023 ELMO Shortlist, A1

Tags: Elmo , algebra , Polynomials , functional equation

Find all polynomials $P(x)$ with real coefficients such that for all nonzero real numbers $x$, \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]