Olympiad problems

2024 USEMO, 4

Tags: algebra , polynomial

Find all sequences $a_1$, $a_2$, $\dots$ of nonnegative integers such that for all positive integers $n$, the polynomial \[1+x^{a_1}+x^{a_2}+\dots+x^{a_n}\] has at least one integer root. (Here $x^0=1$.) [i]Kornpholkrit Weraarchakul[/i]

2009 USA Team Selection Test, 8

Tags: modular arithmetic , algebra , polynomial , calculus , number theory

Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i \plus{} j \plus{} k$ is divisible by $ p \minus{} 1$. Suppose that for all integers $ x$, the quantity \[ (x \minus{} a)(x \minus{} b)(x \minus{} c)[(x \minus{} a)^i(x \minus{} b)^j(x \minus{} c)^k \minus{} 1]\] is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p \minus{} 1$. [i]Kiran Kedlaya and Peter Shor.[/i]

2009 BMO TST, 4

Tags: polynomial , algebra

Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $ \forall x>0$.

2024-25 IOQM India, 24

Tags: algebra , polynomial

Consider the set $F$ of all polynomials whose coefficients are in the set of $\{0,1\}$. Let $q(x) = x^3 + x +1$. The number of polynomials $p(x)$ in $F$ of degree $14$ such that the product $p(x)q(x)$ is also in $F$ is:

2022 Turkey Team Selection Test, 6

Tags: algebra , polynomial , integer polynomial , number theory

For a polynomial $P(x)$ with integer coefficients and a prime $p$, if there is no $n \in \mathbb{Z}$ such that $p|P(n)$, we say that polynomial $P$ [i]excludes[/i] $p$. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?

2005 Kazakhstan National Olympiad, 4

Tags: polynomial , algebra

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

1995 Taiwan National Olympiad, 1

Tags: inequalities , polynomial , algebra

Let $P(x)=a_{0}+a_{1}x+...+a_{n}x^{n}\in\mathbb{C}[x]$ , where $a_{n}=1$. The roots of $P(x)$ are $b_{1},b_{2},...,b_{n}$, where $|b_{1}|,|b_{2}|,...,|b_{j}|>1$ and $|b_{j+1}|,...,|b_{n}|\leq 1$. Prove that $\prod_{i=1}^{j}|b_{i}|\leq\sqrt{|a_{0}|^{2}+|a_{1}|^{2}+...+|a_{n}|^{2}}$.

2014-2015 SDML (High School), 7

Tags: algebra , polynomial

Let $a$, $b$, and $c$ be the roots of the polynomial $$x^3+4x^2-7x-1.$$ Which of the following has roots $ab$, $bc$, and $ac$? $\text{(A) }x^3-4x^2+7x-1\qquad\text{(B) }x^3-7x^2+4x-1\qquad\text{(C) }x^3+7x^2-4x-1\qquad\text{(D) }x^3-4x^2+7x+1\qquad\text{(E) }x^3+7x^2-4x+1$

1996 Baltic Way, 11

Tags: polynomial , algebra

Real numbers $x_1,x_2,\ldots ,x_{1996}$ have the following property: For any polynomial $W$ of degree $2$ at least three of the numbers $W(x_1),W(x_2),\ldots ,W(x_{1996})$ are equal. Prove that at least three of the numbers $x_1,x_2,\ldots ,x_{1996}$ are equal.

2002 Iran MO (3rd Round), 23

Tags: algebra , polynomial , analytic geometry , number theory

Find all polynomials $p$ with real coefficients that if for a real $a$,$p(a)$ is integer then $a$ is integer.

1979 IMO Longlists, 42

Tags: quadratic , polynomial , algebra

Let a quadratic polynomial $g(x) = ax^2 + bx + c$ be given and an integer $n \ge 1$. Prove that there exists at most one polynomial $f(x)$ of $n$th degree such that $f(g(x)) = g(f(x)).$

1996 IMO Shortlist, 5

Tags: algebra , polynomial , inequalities , function , maximization

Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then \[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]

2010 JBMO Shortlist, 1

Tags: polynomial , function , algebra

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

2000 China Team Selection Test, 2

Tags: induction , algebra , polynomial , factorial , combinatorics , finite differences

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

2018 Mathematical Talent Reward Programme, SAQ: P 5

Tags: fibonacci sequence , remainder , periodic sequence , polynomial , algebra

[list=1] [*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number. [*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime. [/list]

2017 Taiwan TST Round 1, 1

Tags: algebra , polynomial , functional equation

Find all polynomials $P$ with real coefficients which satisfy \[P(x)P(x+1)=P(x^2-x+3) \quad \forall x \in \mathbb{R}\]

2021 Stars of Mathematics, 2

Tags: polynomial , algebra , pell equation

Let $n{}$ be a positive integer. Show that there exists a polynomial $f{}$ of degree $n{}$ with integral coefficients such that \[f^2=(x^2-1)g^2+1,\] where $g{}$ is a polynomial with integral coefficients.

2009 Princeton University Math Competition, 2

Tags: algebra , polynomial , quadratic

Given that $P(x)$ is the least degree polynomial with rational coefficients such that \[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.

2016 IFYM, Sozopol, 2

Tags: polynomial , prime number , number theory , algebra

We are given a polynomial $f(x)=x^6-11x^4+36x^2-36$. Prove that for an arbitrary prime number $p$, $f(x)\equiv 0\pmod{p}$ has a solution.

2021 Iran Team Selection Test, 4

Tags: number theory , polynomial , prime number , function , algebra , combi

Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

2011 IFYM, Sozopol, 1

Tags: polynomial , floor function , algebra

Let $n$ be a positive integer. Find the number of all polynomials $P$ with coefficients from the set $\{0,1,2,3\}$ and for which $P(2)=n$.

1987 IMO Longlists, 30

Tags: vector , algebra , polynomial , geometry

Consider the regular $1987$-gon $A_1A_2 . . . A_{1987}$ with center $O$. Show that the sum of vectors belonging to any proper subset of $M = \{OA_j | j = 1, 2, . . . , 1987\}$ is nonzero.

2007 Irish Math Olympiad, 5

Tags: quadratic , polynomial , algebra

Suppose that $ a$ and $ b$ are real numbers such that the quadratic polynomial $ f(x)\equal{}x^2\plus{}ax\plus{}b$ has no nonnegative real roots. Prove that there exist two polynomials $ g,h$ whose coefficients are nonnegative real numbers such that: $ f(x)\equal{}\frac{g(x)}{h(x)}$ for all real numbers $ x$.

2011 AIME Problems, 7

Tags: modular arithmetic , algebra , polynomial , function , number theory

Find the number of positive integers $m$ for which there exist nonnegative integers $x_0,x_1,\ldots,x_{2011}$ such that \[ m^{x_0}=\sum_{k=1}^{2011}m^{x_k}. \]

2019 Hanoi Open Mathematics Competitions, 12

Tags: polynomial , algebra , combinatorics , integer

Given an expression $x^2 + ax + b$ where $a,b$ are integer coefficients. At any step, one can change the expression by adding either $1$ or $-1$ to only one of the two coefficients $a, b$. a) Suppose that the initial expression has $a =-7$ and $b = 19$. Show your modification steps to obtain a new expression that has zero value at some integer value of $x$. b) Starting from the initial expression as above, one gets the expression $x^2 - 17x + 9$ after $m$ modification steps. Prove that at a certain step $k$ with $k < m$, the obtained expression has zero value at some integer value of $x$.