Olympiad problems

I Soros Olympiad 1994-95 (Rus + Ukr), 9.4

The natural numbers $X$ and $Y$ are obtained from each other by permuting the digits. Prove that the sums of the digits of the numbers $5X$ and $5Y$ coincide.

2010 Tournament Of Towns, 1

Tags: number theory

Each of six fruit baskets contains pears, plums and apples. The number of plums in each basket equals the total number of apples in all other baskets combined while the number of apples in each basket equals the total number of pears in all other baskets combined. Prove that the total number of fruits is a multiple of $31$.

2024 IMO, 2

Tags: number theory

Determine all pairs $(a,b)$ of positive integers for which there exist positive integers $g$ and $N$ such that $$\gcd (a^n+b,b^n+a)=g$$ holds for all integers $n\geqslant N.$ (Note that $\gcd(x, y)$ denotes the greatest common divisor of integers $x$ and $y.$) [i]Proposed by Valentio Iverson, Indonesia[/i]

2024-IMOC, N8

Tags: number theory

Find all integers $(a,b)$ satisfying: there is an integer $k>1$ such that $$a^k+b^k-1\ |\ a^n+b^n-1$$ holds for all integer $n\geq k$ (we define that $0|0$)

2009 China Northern MO, 3

Tags: number theory , divide

Given $26$ different positive integers , in any six numbers of the $26$ integers , there are at least two numbers , one can be devided by another. Then prove : There exists six numbers , one of them can be devided by the other five numbers .

2022 Switzerland - Final Round, 7

Tags: number theory , even , prime factorization , perfect number

Let $n > 6$ be a perfect number. Let $p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}$ be the prime factorisation of $n$, where we assume that $p_1 < p_2 <...< p_k$ and $a_i > 0$ for all $ i = 1,...,k$. Prove that $a_1$ is even. Remark: An integer $n \ge 2$ is called a perfect number if the sum of its positive divisors, excluding $ n$ itself, is equal to $n$. For example, $6$ is perfect, as its positive divisors are $\{1, 2, 3, 6\}$ and $1+2+3=6$.

2016 PUMaC Number Theory A, 4

Tags: number theory

Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \le n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.

2020 Bulgaria Team Selection Test, 2

Tags: number theory , algebra

Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$

2015 Middle European Mathematical Olympiad, 7

Tags: factorial , diophantine equation , number theory , exponential equation

Find all pairs of positive integers $(a,b)$ such that $$a!+b!=a^b + b^a.$$

1983 IMO Shortlist, 7

Tags: number theory , sequence , recurrence relation , algebra

Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and \[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\] Show that for each positive integer $n$, $a_n$ is a positive integer.

2005 Slovenia National Olympiad, Problem 4

Tags: number theory

The friends Alex, Ben, and Charles prepared a lot of labels and wrote one of the numbers $2,3,4,5,6,7,8$ on each label. Then Mary joined them and glued one label onto the forehead of each friend. Of course, each of the friends can see the labels on the others’ foreheads, but not the one on his own forehead. Mary told them: ”The numbers on your foreheads are not all distinct, and their product is a perfect square.” Can any of the friends find out the number on his forehead?

2010 Contests, 2

Tags: number theory

Find all prime numbers $p, q, r$ such that \[15p+7pq+qr=pqr.\]

1991 French Mathematical Olympiad, Problem 4

Tags: number theory , set

Let $p$ be a nonnegative integer and let $n=2^p$. Consider all subsets $A$ of the set $\{1,2,\ldots,n\}$ with the property that, whenever $x\in A$, $2x\notin A$. Find the maximum number of elements that such a set $A$ can have.

2001 JBMO ShortLists, 4

Tags: quadratic , algebra , number theory

The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.

1980 IMO Longlists, 9

Tags: pascal triangle , modular arithmetic , number theory , representation

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

2014 USA Team Selection Test, 2

Tags: quadratic residue , quadratic , induction , modular arithmetic , number theory

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

2010 Saudi Arabia IMO TST, 3

Tags: perfect cube , geometry , 3d geometry , number theory

Find all primes $p$ for which $p^2 - p + 1$ is a perfect cube.

2001 China Team Selection Test, 3

Tags: algebra , number theory

Given $a$, $b$ are positive integers greater than $1$, and for every positive integer $n$, $b^{n}-1$ divides $a^{n}-1$. Define the polynomial $p_{n}(x)$ as follows: $p_0{x}=-1$, $p_{n+1}(x)=b^{n+1}(x-1)p_{n}(bx)-a(b^{n+1}-1)p_{n}(x)$, for $n \ge 0$. Prove that there exist integers $C$ and positive integer $k$ such that $p_{k}(x)=Cx^k$.

2022 Azerbaijan National Mathematical Olympiad, 3

Tags: diophantine equation , diophantine , number theory

Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

2001 Singapore Team Selection Test, 3

Tags: lcm , least common multiple , number theory

Let $L(n)$ denote the least common multiple of $\{1, 2 . . . , n\}$. (i) Prove that there exists a positive integer $k$ such that $L(k) = L(k + 1) = ... = L(k + 2000)$. (ii) Find all $m$ such that $L(m + i) \ne L(m + i + 1)$ for all $i = 0, 1, 2$.

2001 Regional Competition For Advanced Students, 1

Tags: digit , power , number theory , sum of powers , sum

Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

2013 VTRMC, Problem 4

Tags: number theory , algebra

A positive integer $n$ is called special if it can be represented in the form $n=\frac{x^2+y^2}{u^2+v^2}$, for some positive integers $x,y,u,v$. Prove that (a) $25$ is special; (b) $2014$ is not special; (c) $2015$ is not special.

2025 Macedonian TST, Problem 6

Tags: number theory

Let $n>2$ be an even integer, and let $V$ be an arbitrary set of $8$ distinct integers. Define \[ E(V,n) \;=\; \bigl\{(u,v)\in V\times V : u < v,\ u+v = n^k\text{ for some }k\in\mathbb{N}\bigr\}. \] For each even $n>2$, determine the maximum possible size of the set $E(V,n)$.

2024 Kyiv City MO Round 1, Problem 5

Tags: number theory , divisor

Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold: $$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$ [i]Proposed by Mykyta Kharin[/i]

1994 Baltic Way, 10

Tags: absolute value , number theory

How many positive integers satisfy the following three conditions: a) All digits of the number are from the set $\{1,2,3,4,5\}$; b) The absolute value of the difference between any two consecutive digits is $1$; c) The integer has $1994$ digits?