Olympiad problems

1980 Canada National Olympiad, 4

A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$.

2024 pOMA, 5

Tags: polynomial , algebra

Prove that there do not exist positive integers $a,b,c$ such that the polynomial \[ P(x) = x^3 - 2^ax^2 + 3^bx - 6^c \] has three integer roots.

1966 IMO Shortlist, 35

Tags: irrational number , polynomial , algebra , root

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

2010 Contests, 4

Tags: algebra , combinatorics , polynomial , linear algebra , modular arithmetic , matrix

Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

2011 German National Olympiad, 6

Tags: prime , number theory , polynomial , sequence , algebra

Let $p>2$ be a prime. Define a sequence $(Q_{n}(x))$ of polynomials such that $Q_{0}(x)=1, Q_{1}(x)=x$ and $Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)$ for $n\geq 1.$ Prove that $Q_{p}(x)-x^p $ is divisible by $p$ for all integers $x.$

2009 Princeton University Math Competition, 2

Tags: algebra , polynomial

Let $p(x)$ be the polynomial with leading coefficent 1 and rational coefficents, such that \[p\left(\sqrt{3 + \sqrt{3 + \sqrt{3 + \ldots}}}\right) = 0,\] and with the least degree among all such polynomials. Find $p(5)$.

2001 Korea Junior Math Olympiad, 2

Tags: algebra , relatively prime , number theory , polynomial

$n$ is a product of some two consecutive primes. $s(n)$ denotes the sum of the divisors of $n$ and $p(n)$ denotes the number of relatively prime positive integers not exceeding $n$. Express $s(n)p(n)$ as a polynomial of $n$.

2020 Silk Road, 3

Tags: algebra , polynomial

A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called [i]powerful[/i] if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and [i]non-increasing[/i] , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $. Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is [i]powerful[/i] for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is [i]nonincreasing[/i].

2006 Iran Team Selection Test, 4

Tags: number theory , algebra , divisibility tests , polynomial

Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

2005 IMC, 4

Tags: polynomial , algebra

4) find all polynom with coeffs a permutation of $[1,...,n]$ and all roots rational

2023 Iran MO (3rd Round), 1

Tags: algebra , polynomial

Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n. We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.

2010 Today's Calculation Of Integral, 586

Tags: calculus , limit , integration , algebra , polynomial , calculus computations , trigonometry

Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.

2014 NIMO Problems, 3

Tags: modular arithmetic , algebra , polynomial , complex number

Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is divided by $1000$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in a set $X$.) [i]Proposed by David Altizio[/i]

2024 USA TSTST, 2

Tags: continued fraction , polynomial , algebra

Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and \[ 1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 + (p-1)x}}}=\frac{P(x)}{Q(x)}. \] Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are [i]not[/i] divisible by $p$. [i]Andrew Gu[/i]

PEN S Problems, 5

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

Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.

2007 All-Russian Olympiad, 6

Tags: polynomial , integration , calculus , quadratic , algebra

Do there exist non-zero reals $a$, $b$, $c$ such that, for any $n>3$, there exists a polynomial $P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c$, which has exactly $n$ (not necessary distinct) integral roots? [i]N. Agakhanov, I. Bogdanov[/i]

2014 Greece National Olympiad, 1

Tags: polynomial , quadratic , algebra

Find all the polynomials with real coefficients which satisfy $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ for all $x\in \mathbb{R}$.

2003 All-Russian Olympiad, 1

Tags: algebra , polynomial , vieta , area of a triangle , heron's formula , geometry

The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

2011 Iran MO (3rd Round), 1

Tags: inequalities , polynomial , algebra

We define the recursive polynomial $T_n(x)$ as follows: $T_0(x)=1$ $T_1(x)=x$ $T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$ $\forall n \in \mathbb N$. [b]a)[/b] find $T_2(x),T_3(x),T_4(x)$ and $T_5(x)$. [b]b)[/b] find all the roots of the polynomial $T_n(x)$ $\forall n \in \mathbb N$. [i]Proposed by Morteza Saghafian[/i]

2004 IMO Shortlist, 4

Tags: functional equation , polynomial , algebra

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations \[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

2012 Pre - Vietnam Mathematical Olympiad, 2

Tags: polynomial , algebra

Let $(a_n)$ defined by: $a_0=1, \; a_1=p, \; a_2=p(p-1)$, $a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}$. Knowing that (i) $a_n>0, \; \forall n \in \mathbb{N}$. (ii) $a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0$. Prove that $|p-1| \ge 2$.

2006 MOP Homework, 4

Tags: system of equations , algebra , inequalities , function , polynomial

Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.

2015 Belarus Team Selection Test, 4

Tags: polynomial , algebra

Find all pairs of polynomials $p(x),q(x)\in R[x]$ satisfying the equality $p(x^2)=p(x)q(1-x)+p(1-x)q(x)$ for all real $x$. I.Voronovich

2013 VJIMC, Problem 3

Tags: polynomial , set , number theory

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 $\max(\deg f,c)$.

2019 IOM, 6

Tags: polynomial , algebra

Let $p$ be a prime and let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Assume that the numbers $f(1),f(2),\dots,f(p)$ leave exactly $k$ distinct remainders when divided by $p$, and $1<k<p$. Prove that \[ \frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .\] [i] Dániel Domán, Gauls Károlyi, and Emil Kiss [/i]