Olympiad problems

2017 Germany, Landesrunde - Grade 11/12, 1

Solve the equation \[ x^5+x^4+x^3+x^2=x+1 \] in $\mathbb{R}$.

2019 SG Originals, Q7

Tags: polynomial , number theory

Let $n$ be a natural number. A sequence is $k-$complete if it contains all residues modulo $n^k$. Let $Q(x)$ be a polynomial with integer coefficients. For $k\ge 2$, define $Q^k(x)=Q(Q^{k-1}(x))$, where $Q^1(x)=Q(x)$. Show that if $$0,Q(0),Q^2(0),Q^3(0),\ldots $$is $2018-$complete, then it is $k-$complete for all positive integers $k$. [i]Proposed by Ma Zhao Yu[/i]

2025 India STEMS Category A, 2

Tags: algebra , polynomial

Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$. [i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]

2007 Hong kong National Olympiad, 2

Tags: algebra , polynomial , modular arithmetic , number theory

is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

2002 Austrian-Polish Competition, 5

Tags: polynomial , modular arithmetic , algebra

Let $A$ be the set $\{2,7,11,13\}$. A polynomial $f$ with integer coefficients possesses the following property: for each integer $n$ there exists $p \in A$ such that $p|f(n)$. Prove that there exists $p \in A$ such that $p|f(n)$ for all integers $n$.

2012 Bulgaria National Olympiad, 3

Tags: polynomial , algebra

We are given a real number $a$, not equal to $0$ or $1$. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation: \[*x^4+*x^3+*x^2+*x^1+*=0\] with a number of the type $a^n$, where $n$ is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of $a$) who has a winning strategy

2011 VJIMC, Problem 4

Tags: polynomial , fe , functional equation , irreducibility , algebra

Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.

2015 Iran MO (3rd round), 4

Tags: algebra , polynomial , complex number

$p(x)\in \mathbb{C}[x]$ is a polynomial such that: $\forall z\in \mathbb{C}, |z|=1\Longrightarrow p(z)\in \mathbb{R}$ Prove that $p(x)$ is constant.

2015 Nordic, 3

Tags: number theory , algebra , polynomial

Let $n > 1$ and $p(x)=x^n+a_{n-1}x^{n-1} +...+a_0$ be a polynomial with $n$ real roots (counted with multiplicity). Let the polynomial $q$ be defined by $$q(x) = \prod_{j=1}^{2015} p(x + j)$$. We know that $p(2015) = 2015$. Prove that $q$ has at least $1970$ different roots $r_1, ..., r_{1970}$ such that $|r_j| < 2015$ for all $ j = 1, ..., 1970$.

2006 Purple Comet Problems, 11

Tags: polynomial , algebra

Consider the polynomials \begin{align*}P(x) &= (x + \sqrt{2})(x^2 - 2x + 2)\\Q(x) &= (x - \sqrt{2})(x^2 + 2x + 2)\\R(x) &= (x^2 + 2)(x^8 + 16).\end{align*} Find the coefficient of $x^4$ in $P(x)\cdot Q(x)\cdot R(x)$.

KoMaL A Problems 2020/2021, A. 789

Tags: algebra , polynomial , absolute value

Let $p(x) = a_{21} x^{21} + a_{20} x^{20} + \dots + a_1 x + 1$ be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than $1/3$, and all the coefficients of $p(x)$ are lying in the interval $[-2019a,2019a]$ for some positive integer $a$. Prove that if this polynomial is reducible in $\mathbb{Z}[x]$, then the coefficients of one of its factors are less than $a$. [i]Submitted by Navid Safaei, Tehran, Iran[/i]

2019 Brazil Undergrad MO, 1

Tags: matrices , linear algebra , polynomial

Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$ with rational entries and size $ 2019 $ such that: a) $ A ^ 3 + 6A ^ 2-2I = 0 $? b) $ A ^ 4 + 6A ^ 3-2I = 0 $?

II Soros Olympiad 1995 - 96 (Russia), 11.4

Tags: algebra , polynomial

Prove that the equation $x^6 - 100x+1 = 0$ has two roots, and both of these roots are positive. a) Find the first non-zero digit in the decimal notation of the lesser root of this equation. b) Find the first two non-zero digits in the decimal notation of the lesser root of this equation.

2019 PUMaC Algebra A, 3

Tags: quadratic , polynomial , algebra

Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.

2014 Harvard-MIT Mathematics Tournament, 10

Tags: algebra , polynomial , function

For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$.

2002 Italy TST, 3

Tags: algebra , polynomial , vieta , relatively prime , number theory

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

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]

2002 IMC, 1

Tags: parabola , quadratic , algebra , polynomial , conic , geometric transformation , geometry

A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co\"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$-axis. Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$.

1997 Baltic Way, 7

Tags: algebra , polynomial , number theory

Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.

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.

2017 All-Russian Olympiad, 2

Tags: quadratic , number theory , algebra , polynomial

$a,b,c$ - different natural numbers. Can we build quadratic polynomial $P(x)=kx^2+lx+m$, with $k,l,m$ are integer, $k>0$ that for some integer points it get values $a^3,b^3,c^3$ ?

2024 Kosovo EGMO Team Selection Test, P2

Tags: polynomial , algebra

Let $n$ be a natural number and the polynomial, $P(x)=x^n+n$. $(a)$ Is it possible that for some odd number $n$ , the polynomial $P(x)$ is composite for all natural numbers $x$. $(b)$ Is it possible that for some even number $n$ , the polynomial $P(x)$ is composite for all natural numbers $x$. Reason your answers.

2007 USAMO, 5

Tags: induction , symmetry , algebra , polynomial

Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

2006 MOP Homework, 6

Tags: polynomial , root , algebra

Let $n$ be an integer greater than $3$. Prove that all the roots of the polynomial $P(x) = x^n - 5x^{n-1} + 12x^{n-2}- 15x^{n-3} + a_{n-4}x^{n-4} +...+ a_0$ cannot be both real and positive.

2008 ITest, 100

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

Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$. It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$. Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$.