Olympiad problems

2013 Saudi Arabia IMO TST, 4

Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.

2008 China Northern MO, 5

Tags: polynomial , algebra , number theory

Assume $n$ is a positive integer and integer $a$ is the root of the equation $$x^4+3ax^2+2ax-2\times 3^n=0.$$ Find all $n$ and $ a$ that satisfy the conditions.

1987 IMO Longlists, 12

Tags: algebra , polynomial , bijection , bijective function

Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers? [i]Proposed by Finland.[/i]

2013 USAMTS Problems, 3

Tags: algebra , function , polynomial

For each positive integer $n\ge2$, find a polynomial $P_n(x)$ with rational coefficients such that $\displaystyle P_n(\sqrt[n]2)=\frac1{1+\sqrt[n]2}$. (Note that $\sqrt[n]2$ denotes the positive $n^\text{th}$ root of $2$.)

1998 India National Olympiad, 5

Tags: inequalities , algebra , polynomial , quadratic , sum of roots , area of a triangle

Suppose $a,b,c$ are three rela numbers such that the quadratic equation \[ x^2 - (a +b +c )x + (ab +bc +ca) = 0 \] has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that (i) The numbers $a,b,c$ are all positive. (ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.

2008 Baltic Way, 4

Tags: algebra , polynomial

The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6\le P(x)\le 4$ or $6\le P(x)\le 16$.

1998 IMC, 2

Tags: polynomial , inequalities , algebra

$S$ ist the set of all cubic polynomials $f$ with $|f(\pm 1)| \leq 1$ and $|f(\pm \frac{1}{2})| \leq 1$. Find $\sup_{f \in S} \max_{-1 \leq x \leq 1} |f''(x)|$ and all members of $f$ which give equality.

2021 Tuymaada Olympiad, 8

Tags: algebra , polynomial , rational function , quadratic

In a sequence $P_n$ of quadratic trinomials each trinomial, starting with the third, is the sum of the two preceding trinomials. The first two trinomials do not have common roots. Is it possible that $P_n$ has an integral root for each $n$?

2019 Romania Team Selection Test, 2

Tags: greatest common divisor , algebra , number theory , modular arithmetic , polynomial

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

1994 Dutch Mathematical Olympiad, 5

Tags: algebra , polynomial , inequalities

Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2\plus{}bx\plus{}c| \le 1$ for all $ x \in [\minus{}1,1]$. Prove that $ |cx^2\plus{}bx\plus{}a| \le 2$ for all $ x \in [\minus{}1,1]$.

1988 Bundeswettbewerb Mathematik, 3

Tags: algebra , polynomial , trigonometry , geometry , parallelogram

Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

2023 VN Math Olympiad For High School Students, Problem 6

Tags: polynomial , algebra , prime number , number theory

Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$ a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number. b) $x^{2^n}+1,$ with $n$ is a positive integer.

2008 Harvard-MIT Mathematics Tournament, 13

Tags: algebra , factorial , polynomial

Let $ P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that \[ P(0) \equal{} 2007, P(1) \equal{} 2006, P(2) \equal{} 2005, \dots, P(2007) \equal{} 0. \]Determine the value of $ P(2008)$. You may use factorials in your answer.

2007 Moldova National Olympiad, 12.1

Tags: algebra , polynomial

For $a\in C^{*}$ find all $n\in N$ such that $X^{2}(X^{2}-aX+a^{2})^{2}$ divides $(X^{2}+a^{2})^{n}-X^{2n}-a^{2n}$

1954 AMC 12/AHSME, 41

Tags: algebra , polynomial , vieta

The sum of all the roots of $ 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0$ is: $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \minus{}8 \qquad \textbf{(D)}\ \minus{}2 \qquad \textbf{(E)}\ 0$

2011 All-Russian Olympiad, 4

Tags: combinatorics , symmetry , polynomial , rectangle , algebra , modular arithmetic

There are some counters in some cells of $100\times 100$ board. Call a cell [i]nice[/i] if there are an even number of counters in adjacent cells. Can exactly one cell be [i]nice[/i]? [i]K. Knop[/i]

2019 Belarusian National Olympiad, 11.2

Tags: algebra , polynomial

The polynomial $$ Q(x_1,x_2,\ldots,x_4)=4(x_1^2+x_2^2+x_3^2+x_4^2)-(x_1+x_2+x_3+x_4)^2 $$ is represented as the sum of squares of four polynomials of four variables with integer coefficients. [b]a)[/b] Find at least one such representation [b]b)[/b] Prove that for any such representation at least one of the four polynomials isidentically zero. [i](A. Yuran)[/i]

2006 IberoAmerican Olympiad For University Students, 4

Tags: real analysis , algebra , integration , calculus , polynomial , function

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval \[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\] such that \[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\] for all polynomials $f$ with real coefficients and degree less than $n$.

2017 European Mathematical Cup, 4

Tags: polynomial , algebra

Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$ is a square of an integer for all nonnegative integers $n, m$. [i]Remark:[/i] For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$. Proposed by Adrian Beker.

2016 AIME Problems, 6

Tags: algebra , polynomial

For polynomial $P(x)=1-\frac{1}{3}x+\frac{1}{6}x^2$, define \[ Q(x) = P(x)P(x^3)P(x^5)P(x^7)P(x^9) = \sum\limits_{i=0}^{50}a_ix^i. \] Then $\sum\limits_{i=0}^{50}|a_i|=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2005 Kazakhstan National Olympiad, 4

Tags: algebra , polynomial

Find all polynomials $ P(x)$ with real coefficients such that for every positive integer $ n$ there exists a rational $ r$ with $ P(r)=n$.

2010 Contests, 1

Tags: algebra , polynomial

Find the sum of the coefficients of the polynomial $(63x-61)^4$.

2007 Ukraine Team Selection Test, 3

Tags: logarithm , algebra , polynomial

It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.

2005 Tuymaada Olympiad, 2

Tags: algebra , polynomial

Six members of the team of Fatalia for the International Mathematical Olympiad are selected from $13$ candidates. At the TST the candidates got $a_1,a_2, \ldots, a_{13}$ points with $a_i \neq a_j$ if $i \neq j$. The team leader has already $6$ candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial $P(x)$ and finds the creative potential of each candidate by the formula $c_i = P(a_i)$. For what minimum $n$ can he always find a polynomial $P(x)$ of degree not exceeding $n$ such that the creative potential of all $6$ candidates is strictly more than that of the $7$ others? [i]Proposed by F. Petrov, K. Sukhov[/i]

2020 BMT Fall, 2

Tags: polynomial , algebra , sum of roots

Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$. Given that $\frac{a}{b}+\frac{b}{a}=98$, compute $\sqrt c$.