Olympiad problems

2000 India National Olympiad, 5

Tags: polynomial , algebra

Let $a,b,c$ be three real numbers such that $1 \geq a \geq b \geq c \geq 0$. prove that if $\lambda$ is a root of the cubic equation $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq 1.$

1999 Moldova Team Selection Test, 9

Tags: polynomial , algebra

Let $P(X)$ be a nonconstant polynomial with real coefficients such that for every rational number $q{}$ the equation $P(X)=q$ has no irrational solutions. Show that $P(X)$ is a first degree polynomial.

2006 IMC, 4

Tags: algebra , function , rational function , polynomial

Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers n. Prove that f is a polynomial.

2008 Postal Coaching, 3

Tags: algebra , polynomial

Find all real polynomials $P(x, y)$ such that $P(x+y, x-y) = 2P(x, y)$, for all $x, y$ in $R$.

2009 Indonesia TST, 2

Tags: algebra , induction , polynomial

Let $ f(x)\equal{}a_{2n}x^{2n}\plus{}a_{2n\minus{}1}x^{2n\minus{}1}\plus{}\cdots\plus{}a_1x\plus{}a_0$, with $ a_i\equal{}a_{2n\minus{}1}$ for all $ i\equal{}1,2,\ldots,n$ and $ a_{2n}\ne0$. Prove that there exists a polynomial $ g(x)$ of degree $ n$ such that $ g\left(x\plus{}\frac1x\right)x^n\equal{}f(x)$.

2018 Azerbaijan Senior NMO, 2

Tags: algebra , polynomial

$P(x)$ is a fifth degree polynomial. $P(2018)=1$, $P(2019)=2$ $P(2020)=3$, $P(2021)=4$, $P(2022)=5$. $P(2017)=?$

2010 Indonesia TST, 1

Tags: number theory , polynomial , algebra

Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. [i]Nanang Susyanto, Jogjakarta[/i]

2007 All-Russian Olympiad Regional Round, 11.2

Tags: algebra , quadratic , polynomial

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]$.

1955 Polish MO Finals, 1

Tags: algebra , polynomial

What conditions must the real numbers $ a $, $ b $, and $ c $ satisfy so that the equation $$ x^3 + ax^2 + bx + c = 0$$ has three distinct real roots forming a geometric progression?

2019 Peru Cono Sur TST, P6

Tags: calculus , algebra , polynomial

Two polynomials of the same degree $A(x)=a_nx^n+ \cdots + a_1x+a_0$ and $B(x)=b_nx^n+\cdots+b_1x+b_0$ are called [i]friends[/i] is the coefficients $b_0,b_1, \ldots, b_n$ are a permutation of the coefficients $a_0,a_1, \ldots, a_n$. $P(x)$ and $Q(x)$ be two friendly polynomials with integer coefficients. If $P(16)=3^{2020}$, the smallest possible value of $|Q(3^{2020})|$.

1986 Federal Competition For Advanced Students, P2, 3

Tags: limit , algebra , polynomial

Find all possible values of $ x_0$ and $ x_1$ such that the sequence defined by: $ x_{n\plus{}1}\equal{}\frac{x_{n\minus{}1} x_n}{3x_{n\minus{}1}\minus{}2x_n}$ for $ n \ge 1$ contains infinitely many natural numbers.

2021 Ukraine National Mathematical Olympiad, 2

Tags: divisible , polynomial , algebra

Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$. (Oleksii Masalitin)

2012 National Olympiad First Round, 23

Tags: vieta , polynomial , algebra

$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is $ \textbf{(A)}\ 156 \qquad \textbf{(B)}\ 171 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 186 \qquad \textbf{(E)}\ 201$

1991 Arnold's Trivium, 95

Tags: rotation , algebra , geometry , invariant , quadratic , geometric transformation , polynomial

Decompose the space of homogeneous polynomials of degree $5$ in $(x, y, z)$ into irreducible subspaces invariant with respect to the rotation group $SO(3)$.

2016 Postal Coaching, 4

Tags: algebra , polynomial

Let $f$ be a polynomial with real coefficients and suppose $f$ has no nonnegative real root. Prove that there exists a polynomial $h$ with real coefficients such that the coefficients of $fh$ are nonnegative.

1983 Putnam, B6

Tags: polynomial , algebra

Let $ k$ be a positive integer, let $ m\equal{}2^k\plus{}1$, and let $ r\neq 1$ be a complex root of $ z^m\minus{}1\equal{}0$. Prove that there exist polynomials $ P(z)$ and $ Q(z)$ with integer coefficients such that $ (P(r))^2\plus{}(Q(r))^2\equal{}\minus{}1$.

2001 India IMO Training Camp, 2

Tags: algebra , polynomial , number theory

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.

2019 Middle European Mathematical Olympiad, 2

Tags: algebra , root , polynomial

Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$. [i]Proposed by Walther Janous, Austria[/i]

2016 Hanoi Open Mathematics Competitions, 15

Tags: prime , algebra , integer , polynomial

Find all polynomials of degree $3$ with integer coeffcients such that $f(2014) = 2015, f(2015) = 2016$ and $f(2013) - f(2016)$ is a prime number.

2021 Korea National Olympiad, P5

Tags: algebra , sequence , polynomial , inequalities

A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions. $$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$ Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$. Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$ $|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.

2022/2023 Tournament of Towns, P3

Tags: algebra , polynomial , number theory

$P(x)$ is polynomial with degree $n>5$ and integer coefficients have $n$ different integer roots. Prove that $P(x)+3$ have $n$ different real roots.

2014 Brazil Team Selection Test, 2

Tags: number theory , polynomial , prime divisor

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

2012 Indonesia TST, 1

Tags: induction , polynomial , function , algebra

Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$.

2020 Dutch IMO TST, 1

Tags: algebra , polynomial , inequalities

Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different. For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$. Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.

2000 Polish MO Finals, 3

Tags: algebra , polynomial

Show that the only polynomial of odd degree satisfying $p(x^2-1) = p(x)^2 - 1$ for all $x$ is $p(x) = x$