Olympiad problems

2023 Bulgaria JBMO TST, 2

Determine the smallest positive integer $n\geq 2$ for which there exists a positive integer $m$ such that $mn$ divides $m^{2023} + n^{2023} + n$.

1999 IMO Shortlist, 7

Tags: combinatorics , counting , subset , set systems , divisibility , combinatorial inequality

Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$, let $E(T)$ be the set of all $(p-1)$-tuples $(x_1, \ldots ,x_{p-1} )$, where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that \[|E(\{0,1,3\})| \geq |E(\{0,1,2\})|\] with equality if and only if $p = 5$.

2020 Macedonia Additional BMO TST, 3

Tags: number theory , divisibility

Does there exist a set of $2020$ distinct positive whole numbers with the property that the product of any $101$ of them is divisible by the sum of those $101$ numbers?

1968 IMO Shortlist, 21

Tags: divisibility , number theory

Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that \[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]

2016 Bangladesh Mathematical Olympiad, 2

Tags: modular arithmetic , divisibility , number theory

(a) How many positive integer factors does $6000$ have? (b) How many positive integer factors of $6000$ are not perfect squares?

2000 Moldova National Olympiad, Problem 1

Tags: divisibility , determinant , number theory

Let $1=d_1<d_2<\ldots<d_{2m}=n$ be the divisors of a positive integer $n$, where $n$ is not a perfect square. Consider the determinant $$D=\begin{vmatrix}n+d_1&n&\ldots&n\\n&n+d_2&\ldots&n\\\ldots&\ldots&&\ldots\\n&n&\ldots&n+d_{2m}\end{vmatrix}.$$ (a) Prove that $n^m$ divides $D$. (b) Prove that $1+d_1+d_2+\ldots+d_{2m}$ divides $D$.

2017 Germany Team Selection Test, 3

Tags: function , number theory , divisibility

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2009 Serbia Team Selection Test, 1

Tags: combinatorics , permutation , divisibility

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

1999 IMO Shortlist, 3

Tags: number theory , integer sequence , divisibility , sequence , modular arithmetic

Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

1998 IMO Shortlist, 3

Tags: combinatorics , pigeonhole principle , divisibility , number theory , modular arithmetic

Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.

2023 Grosman Mathematical Olympiad, 1

Tags: divisibility , arithmetic sequence , number theory

An arithmetic progression of natural numbers of length $10$ and with difference $11$ is given. Prove that the product of the numbers in this progression is divisible by $10!$.

2023 Ukraine National Mathematical Olympiad, 9.2

Tags: number theory , divisibility

Positive integers $a_1, a_2, \ldots, a_{101}$ are such that $a_i+1$ is divisible by $a_{i+1}$ for all $1 \le i \le 101$, where $a_{102} = a_1$. What is the largest possible value of $\max(a_1, a_2, \ldots, a_{101})$? [i]Proposed by Oleksiy Masalitin[/i]

2010 Belarus Team Selection Test, 3.2

Tags: divisibility , number theory

Prove that there exists a positive integer $n$ such that $n^6 + 31n^4 - 900\vdots 2009 \cdot 2010 \cdot 2011$. (I. Losev, I. Voronovich)

1993 Mexico National Olympiad, 6

Tags: number theory , prime , divisibility

$p$ is an odd prime. Show that $p$ divides $n(n+1)(n+2)(n+3) + 1$ for some integer $n$ iff $p$ divides $m^2 - 5$ for some integer $m$.

2015 Danube Mathematical Competition, 3

Tags: divisibility , rmn

Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.

Russian TST 2016, P1

Tags: number theory , divisibility

Let $a{}$ and $b{}$ be natural numbers greater than one. Let $n{}$ be a natural number for which $a\mid 2^n-1$ and $b\mid 2^n+1$. Prove that there is no natural $k{}$ such that $a\mid 2^k+1$ and $b\mid 2^k-1$.

2021 Bolivian Cono Sur TST, 1

Tags: number theory , divisibility

Find the sum of all positive integers $n$ such that $$\frac{n+11}{\sqrt{n-1}}$$ is an integer.

2016 IMO Shortlist, N6

Tags: divisibility , number theory , function

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2006 Taiwan TST Round 1, 3

Tags: number theory , divisibility , chinese remainder theorem , modular arithmetic

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

1969 IMO Shortlist, 25

Tags: divisibility , frobenius , additive number theory , number theory

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

1969 Bulgaria National Olympiad, Problem 1

Tags: number theory , divisibility

Prove that if the sum of $x^5,y^5$ and $z^5$, where $x,y$ and $z$ are integer numbers, is divisible by $25$ then the sum of some two of them is divisible by $25$.

2011 Brazil Team Selection Test, 4

Tags: divisibility , modular arithmetic , number theory

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

2018 Brazil Team Selection Test, 3

Tags: number theory , combinatorics , counting , divisibility

Let $n > 10$ be an odd integer. Determine the number of ways to place the numbers $1, 2, \ldots , n$ around a circle so that each number in the circle divides the sum its two neighbors. (Two configurations such that one can be obtained from the other per rotation are to be counted only once.)

1999 Romania Team Selection Test, 10

Tags: divisibility , modular arithmetic , number theory

Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

2022 Olimphíada, 1

Tags: prime , divisibility

Let $p,q$ prime numbers such that $$p+q \mid p^3-q^3$$ Show that $p=q$.