Olympiad problems

2008 Germany Team Selection Test, 1

Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

2020 AMC 10, 6

Tags: AMC , AMC 12 , 2020 AMC , 2020 AMC 12A , Divisibility , 2020 AMC 10A , AMC 10

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$ $\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$

1969 IMO Longlists, 43

Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist , IMO Longlist

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$

2017 Ukraine Team Selection Test, 2

Tags: number theory , IMO Shortlist , function , Divisibility , Hi

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]

2022 Bundeswettbewerb Mathematik, 1

Tags: number theory , Divisibility , gauss sum

Five squirrels together have a supply of 2022 nuts. On the first day 2 nuts are added, on the second day 4 nuts, on the third day 6 nuts and so on, i.e. on each further day 2 nuts more are added than on the day before. At the end of any day the squirrels divide the stock among themselves. Is it possible that they all receive the same number of nuts and that no nut is left over?

2016 Taiwan TST Round 2, 1

Tags: factorial , number theory , IMO Shortlist , Divisibility

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

2008 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , Divisibility , Extremal combinatorics , Additive combinatorics , IMO Shortlist

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

2023 Romania Team Selection Test, P3

Tags: number theory , factorial , Divisibility

Given a positive integer $a,$ prove that $n!$ is divisible by $n^2 + n + a$ for infinitely many positive integers $n.{}$ [i]Proposed by Andrei Bâra[/i]

2017 JBMO Shortlist, NT2

Tags: number theory , Divisibility , factorial

Determine all positive integers n such that $n^2/ (n - 1)!$

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P4

Tags: number theory , Sets , disjoint subsets , Divisibility

Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$, i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$, such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the elements in $S_2$. [i]Proposed by Viktor Simjanoski[/i]

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?

2023 239 Open Mathematical Olympiad, 2

Tags: number theory , Divisibility

Let $1 < a_1 < a_2 < \cdots < a_N$ be natural numbers. It is known that for any $1\leqslant i\leqslant N$ the product of all these numbers except $a_i$ increased by one, is a multiple of $a_i$. Prove that $a_1\leqslant N$.

2021 China Team Selection Test, 2

Tags: number theory , Number sequence , Divisibility

Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.

2014 India Regional Mathematical Olympiad, 6

Tags: combinatorics , number theory , Divisibility

In the adjacent figure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3$?

1969 IMO Shortlist, 43

Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist , IMO Longlist

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$

2024 Brazil Cono Sur TST, 3

Tags: Hensel s Lemma , number theory , Quadratic Residues , Chinese Remainder Theorem , Divisibility

Find all positive integers $m$ that have some multiple of the form $x^2+5y^2+2024$, with $x$ and $y$ integers.

2014 District Olympiad, 4

Tags: function , algebra , functional equation , Divisibility

Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with the properties: [list=a] [*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $ [*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]

2014 BAMO, 4

Tags: Fibonacci , Sequence , Divisibility , number theory

Let $F_1, F_2, F_3 \cdots$ be the Fibonacci sequence, the sequence of positive integers satisfying $$F_1 =F_2=1$$ and $$F_{n+2} = F_{n+1} + F_n$$ for all $n \ge 1$. Does there exist an $n \ge 1$ such that $F_n$ is divisible by $2014$? Prove your answer.

2016 Bangladesh Mathematical Olympiad, 2

Tags: number theory , modular arithmetic , Divisibility , Bdmo , contests

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

2019 India PRMO, 8

Tags: algebra , number theory , PRMO , Divisibility

Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$. For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer?

2015 Israel National Olympiad, 4

Tags: number theory , Divisibility

Let $k,m,n$ be positive integers such that $n^m$ is divisible by $m^n$, and $m^k$ is divisible by $k^m$. [list=a] [*] Prove that $n^k$ is divisible by $k^n$. [*] Find an example of $k,m,n$ satisfying the above conditions, where all three numbers are distinct and bigger than 1. [/list]

2018 South Africa National Olympiad, 5

Tags: number theory , Sequences , Divisibility

Determine all sequences $a_1, a_2, a_3, \dots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \dots$ and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$.

2017 Germany Team Selection Test, 3

Tags: number theory , IMO Shortlist , function , Divisibility , Hi

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]

2010 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , Divisibility , odd

Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$

1984 IMO Longlists, 43

Tags: number theory , equation , Divisibility , power of 2 , IMO , IMO 1984 , Hi

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 $a=1$.