Olympiad problems

2018 JBMO Shortlist, NT1

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

2022 Turkey Team Selection Test, 1

Tags: modular arithmetic , diophantine equation , number theory

Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

2011 Vietnam Team Selection Test, 4

Tags: floor function , induction , number theory

Let $\langle a_n\rangle_{n\ge 0}$ be a sequence of integers satisfying $a_0=1, a_1=3$ and $a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.$ Prove that $a_n\cdot a_{n+2}-a_{n+1}^2=2^n$ for every natural number $n.$

1988 Polish MO Finals, 2

Tags: induction , pigeonhole principle , modular arithmetic , number theory

The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = a_2 = a_3 = 1$, $a_{n+3} = a_{n+2}a_{n+1} + a_n$. Show that for any positive integer $r$ we can find $s$ such that $a_s$ is a multiple of $r$.

2024 European Mathematical Cup, 1

Tags: graph theory , number theory

We call a pair of distinct numbers $(a, b)$ a [i]binary pair[/i] if $ab+1$ is a power of two. Given a set $S$ of $n$ positive integers, what is the maximum possible numbers of binary pairs in S?

2007 Bulgarian Autumn Math Competition, Problem 12.4

Tags: prime number , number theory , linear recurrence

Let $p$ and $q$ be prime numbers and $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers defined by: \[a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n}\quad\forall n\geq 0\] Find $p$ and $q$ if there exists an integer $k$ such that $a_{3k}=-3$.

1988 Tournament Of Towns, (193) 6

Tags: number theory

Does there exist a natural number which is not a divisor of any natural number whose decimal expression consists of zeros and ones, with no more than $1988$ ones?

2021 Iran Team Selection Test, 3

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

Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root) Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]

2009 Indonesia TST, 1

Tags: algebra , polynomial , quadratic , number theory

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

2024 All-Russian Olympiad, 1

Tags: number theory

Let $p$ and $q$ be different prime numbers. We are given an infinite decreasing arithmetic progression in which each of the numbers $p^{23}, p^{24}, q^{23}$ and $q^{24}$ occurs. Show that the numbers $p$ and $q$ also occur in this progression. [i]Proposed by A. Kuznetsov[/i]

2017 Pan-African Shortlist, N1

Tags: number theory , greatest common divisor , floor function , function , algebra , divisibility theory

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

PEN A Problems, 110

Tags: number theory , relatively prime , divisibility theory

For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$.

2025 Vietnam Team Selection Test, 4

Tags: number theory , binomial

Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.

2024 JBMO TST - Turkey, 4

Tags: number theory

Let $n$ be a positive integer and $d(n)$ is the number of positive integer divisors of $n$. For every two positive integer divisor $x,y$ of $n$, the remainders when $x,y$ divided by $d(n)+1$ are pairwise distinct. Show that either $d(n)+1$ is equal to prime or $4$.

2015 India Regional MathematicaI Olympiad, 6

Tags: number theory , fractional part , integer

Find all real numbers $a$ such that $3 < a < 4$ and $a(a-3\{a\})$ is an integer. (Here $\{a\}$ denotes the fractional part of $a$.)

2003 Croatia Team Selection Test, 1

Tags: perfect square , number theory

Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

2016 Romanian Masters in Mathematic, 3

Tags: number theory

A $\textit{cubic sequence}$ is a sequence of integers given by $a_n =n^3 + bn^2 + cn + d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers. $\textbf{(a)}$ Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are $a_{2015}$ and $a_{2016}$. $\textbf{(b)}$ Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part $\textbf{(a)}$.

2019 Putnam, A5

Tags: number theory

Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x) = \sum_{k=1}^{p-1} a_k x^k$ where $a_k = k^{(p-1)/2}$ mod $p$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.

2016 Latvia Baltic Way TST, 20

Tags: power of prime , number theory , perfect power

For what pairs of natural numbers $(a, b)$ is the expression $$(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)$$ the power of a prime number?

2013 Gulf Math Olympiad, 4

Tags: greatest common divisor , number theory

Let $m,n$ be integers. It is known that there are integers $a,b$ such that $am+bn=1$ if, and only if, the greatest common divisor of $m,n$ is 1. [i]You are not required to prove this[/i]. Now suppose that $p,q$ are different odd primes. In each case determine if there are integers $a,b$ such that $ap+bq=1$ so that the given condition is satisfied: [list] a. $p$ divides $b$ and $q$ divides $a$; b. $p$ divides $a$ and $q$ divides $b$; c. $p$ does not divide $a$ and $q$ does not divide $b$. [/list]

2020 Mediterranean Mathematics Olympiad, 2

Tags: number theory , combinatorics

Let $S$ be a set of $n\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\in S$. [i]Proposed by Gerhard Woeginger, Austria[/i]

2022 Latvia Baltic Way TST, P16

Tags: number theory

Find all triples of positive integers $(a,b,p)$, where $p$ is a prime, such that both $a+b$ and $ab+1$ are some powers of $p$ (not necessarily the same).

2020 South Africa National Olympiad, 1

Tags: number theory , divisor

Find the smallest positive multiple of $20$ with exactly $20$ positive divisors.

2007 Germany Team Selection Test, 3

Tags: number theory , rational , decimal representation

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

2014 HMNT, 1

Tags: number theory

What is the smallest positive integer $n$ which cannot be written in any of the following forms? $\bullet$ $n = 1 + 2 +... + k$ for a positive integer $k$. $\bullet$ $n = p^k$ for a prime number $p$ and integer $k$. $\bullet$ $n = p + 1$ for a prime number $p$.