Olympiad problems

2012 ELMO Shortlist, 1

Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square. [i]David Yang, Alex Zhu.[/i]

2006 MOP Homework, 3

Tags: number theory , greatest common divisor , inequalities

Prove that the following inequality holds with the exception of finitely many positive integers $n$: $\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.

2004 Balkan MO, 2

Tags: number theory , modular arithmetic

Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$.

2012 USA TSTST, 3

Tags: relatively prime , function , greatest common divisor , number theory , arithmetic sequence , modular arithmetic , induction

Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions: (a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime. (b) $n \le f(n) \le n+2012$ for all $n$. Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.

1999 VJIMC, Problem 2

Tags: number theory

Find all natural numbers $n\ge1$ such that the implication $$(11\mid a^n+b^n)\implies(11\mid a\wedge11\mid b)$$holds for any two natural numbers $a$ and $b$.

1984 IMO Longlists, 43

Tags: equation , divisibility , power of 2 , number theory

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

2012 Irish Math Olympiad, 1

Tags: number theory

Let $S(n)$ be the sum of the decimal digits of $n$. For example. $S(2012)=2+0+1+2=5$. Prove that there is no integer $n>0$ for which $n-S(n)=9990$.

2020 IMO, 5

Tags: arithmetic mean , geometric mean , number theory

A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$ does it follow that the numbers on the cards are all equal? [i]Proposed by Oleg Košik, Estonia[/i]

2023 Kyiv City MO Round 1, Problem 4

Tags: number theory

Find all pairs $(m, n)$ of positive integers, for which number $2^n - 13^m$ is a cube of a positive integer. [i]Proposed by Oleksiy Masalitin[/i]

2010 Belarus Team Selection Test, 2.3

Tags: divisibility , number theory

Prove that there are infinitely many positive integers $n$ such that $$3^{(n-2)^{n-1}-1} -1\vdots 17n^2$$ (I. Bliznets)

2023 Poland - Second Round, 1

Tags: divisibility , number theory

Find all positive integers $b$ with the following property: there exists positive integers $a,k,l$ such that $a^k + b^l$ and $a^l + b^k$ are divisible by $b^{k+l}$ where $k \neq l$.

2013 Saudi Arabia GMO TST, 4

Tags: number theory , fibonacci number , remainder

Let $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$, for all positive integer $n$, be the Fibonacci sequence. Prove that for any positive integer $m$ there exist infinitely many positive integers $n$ such that $F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2}$ mod $m$ .

2022 Philippine MO, 3

Tags: algebra , number theory

Call a lattice point [i]visible[/i] if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum \[ \sum_{d \in D} |S_d| \] Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.

2004 France Team Selection Test, 1

Tags: modular arithmetic , number theory , quadratic

If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

2020 Baltic Way, 17

Tags: number theory

For a prime number $p$ and a positive integer $n$, denote by $f(p, n)$ the largest integer $k$ such that $p^k \mid n!$. Let $p$ be a given prime number and let $m$ and $c$ be given positive integers. Prove that there exist infinitely many positive integers $n$ such that $f(p, n) \equiv c \pmod m$.

2008 Hanoi Open Mathematics Competitions, 3

Tags: number theory , diophantine , diophantine equation

Show that the equation $x^2 + 8z = 3 + 2y^2$ has no solutions of positive integers $x, y$ and $z$.

2024 Iran MO (2nd Round), 3

Tags: number theory

Find all natural numbers $x,y>1$and primes $p$ that satisfy $$\frac{x^2-1}{y^2-1}=(p+1)^2. $$

2019 India PRMO, 7

Tags: algebra , decimal representation , number theory

Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m)=20$ and $s(33m)=120$, what is the value of $s(3m)$?

2007 Singapore Team Selection Test, 3

Tags: greatest common divisor , modular arithmetic , analytic geometry , number theory

Let $A,B,C$ be $3$ points on the plane with integral coordinates. Prove that there exists a point $P$ with integral coordinates distinct from $A,B$ and $C$ such that the interiors of the segments $PA,PB$ and $PC$ do not contain points with integral coordinates.

2016 China Western Mathematical Olympiad, 5

Tags: number theory

Prove that there exist infinitely many positive integer triples $(a,b,c)$ such that $a ,b,c$ are pairwise relatively prime ,and $ab+c ,bc+a ,ca+b$ are pairwise relatively prime .

2023 Chile TST Ibero., 1

Tags: number theory

Given a non-negative integer $ n $, determine the values of $ c $ for which the sequence of numbers \[ a_n = 4^n c + \frac{4^n - (-1)^n}{5} \] contains at least one perfect square.

2022 Middle European Mathematical Olympiad, 4

Tags: number theory

Initially, two distinct positive integers $a$ and $b$ are written on a blackboard. At each step, Andrea picks two distinct numbers $x$ and $y$ on the blackboard and writes the number $gcd(x, y) + lcm(x, y)$ on the blackboard as well. Let $n$ be a positive integer. Prove that, regardless of the values of $a$ and $b$, Andrea can perform a finite number of steps such that a multiple of $n$ appears on the blackboard.

2005 Morocco TST, 1

Tags: number theory

Find all the positive primes $p$ for which there exist integers $m,n$ satisfying : $p=m^2+n^2$ and $m^3+n^3-4$ is divisible by $p$.

2024 Ukraine National Mathematical Olympiad, Problem 2

Tags: algebra , number theory , power of 2

You are given a positive integer $n$. Find the smallest positive integer $k$, for which there exist integers $a_1, a_2, \ldots, a_k$, for which the following equality holds: $$2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k$$ [i]Proposed by Mykhailo Shtandenko[/i]

2016 China Team Selection Test, 6

Tags: combinatorics , combinatorial number theory , number theory

Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).