Found problems: 536
1998 IMO Shortlist, 5
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.
2024 Kyiv City MO Round 1, Problem 3
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2024$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2024$ loses. Who wins if every player wants to win?
[i]Proposed by Mykhailo Shtandenko[/i]
1987 IMO Shortlist, 8
(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$
(b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$
[i]Proposed by Hungary.[/i]
2016 Saudi Arabia BMO TST, 1
Let $ a > b > c > d $ be positive integers such that
\begin{align*}
a^2 + ac - c^2 = b^2 + bd - d^2
\end{align*}
Prove that $ ab + cd $ is a composite number.
2015 Harvard-MIT Mathematics Tournament, 4
Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold.
[list]
[*] $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$
[*] There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such that $a_i+a_j-N$ is divisible by $203$.
[/list]
Maryland University HSMC part II, 2023.3
Let $p$ be a prime, and $n > p$ be an integer. Prove that
\[ \binom{n+p-1}{p} - \binom{n}{p} \]
is divisible by $n$.
2011 IMO Shortlist, 6
Let $P(x)$ and $Q(x)$ be two polynomials with integer coefficients, such that no nonconstant polynomial with rational coefficients divides both $P(x)$ and $Q(x).$ Suppose that for every positive integer $n$ the integers $P(n)$ and $Q(n)$ are positive, and $2^{Q(n)}-1$ divides $3^{P(n)}-1.$ Prove that $Q(x)$ is a constant polynomial.
[i]Proposed by Oleksiy Klurman, Ukraine[/i]
2020 IMO Shortlist, N1
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
2006 Germany Team Selection Test, 2
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n \plus{} 1
\]
[i]Proposed by Carlos Caicedo, Colombia[/i]
2015 German National Olympiad, 4
Let $k$ be a positive integer. Define $n_k$ to be the number with decimal representation $70...01$ where there are exactly $k$ zeroes. Prove the following assertions:
a) None of the numbers $n_k$ is divisible by $13$.
b) Infinitely many of the numbers $n_k$ are divisible by $17$.
1984 IMO Longlists, 40
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
1969 IMO Shortlist, 28
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
1997 IMO Shortlist, 14
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
2019 IMEO, 5
Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$
[i]Proposed by Oleksii Masalitin (Ukraine)[/i]
2023 Myanmar IMO Training, 1
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that
$$m+f(n) \mid f(m)^2 - nf(n)$$
for all positive integers $m$ and $n$.
(Here, $f(m)^2$ denotes $\left(f(m)\right)^2$.)
2018 China Team Selection Test, 6
Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions:
(1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$
(2) For all $n\ge M$, $f(n)\ge 0$
(3) For all $n>m>M$, $n-m|f(n)-f(m)$
Show that $a$ is a perfect $r$-th power.
2018 Pan-African Shortlist, N2
A positive integer is called special if its digits can be arranged to form an integer divisible by $4$. How many of the integers from $1$ to $2018$ are special?
2021 Polish Junior MO Finals, 5
Natural numbers $a$, $b$ are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in $a$ and in $b$). Prove that if $a+b=10^{1000}$ then both numbers $a$ and $b$ are divisible by $10$.
1992 IMO Longlists, 31
Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$
2012 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$
2024 Baltic Way, 17
Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?
2018 India Regional Mathematical Olympiad, 5
Find all natural numbers $n$ such that $1+[\sqrt{2n}]~$ divides $2n$.
( For any real number $x$ , $[x]$ denotes the largest integer not exceeding $x$. )
Kvant 2023, M2768
Let $n{}$ be a natural number. The pairwise distinct nonzero integers $a_1,a_2,\ldots,a_n$ have the property that the number \[(k+a_1)(k+a_2)\cdots(k+a_n)\]is divisible by $a_1a_2\cdots a_n$ for any integer $k{}.$ Find the largest possible value of $a_n.$
[i]Proposed by F. Petrov and K. Sukhov[/i]
1993 IMO Shortlist, 3
Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.
2005 Taiwan TST Round 3, 1
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]