Found problems: 545
2015 Bosnia Herzegovina Team Selection Test, 3
Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.
2020 Macedonian Nationаl Olympiad, 1
Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.
1992 IMO Longlists, 14
Integers $a_1, a_2, . . . , a_n$ satisfy $|a_k| = 1$ and
\[ \sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,\]
where $a_{n+j} = a_j$. Prove that $n \neq 1992.$
2022 IMO Shortlist, N4
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
2024 Middle European Mathematical Olympiad, 8
Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that
\[a_ia_{i+1} \mid k-a_i^2\]
for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all
integers $n \ge M$.
2016 Peru IMO TST, 16
Find all pairs $ (m, n)$ of positive integers that have the following property:
For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.
2005 China Team Selection Test, 1
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.
2015 Harvard-MIT Mathematics Tournament, 9
Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.
2018 China Team Selection Test, 5
Given a positive integer $k$, call $n$ [i]good[/i] if among $$\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n}$$ at least $0.99n$ of them are divisible by $k$. Show that exists some positive integer $N$ such that among $1,2,...,N$, there are at least $0.99N$ good numbers.
2013 IFYM, Sozopol, 3
Let $a$ and $b$ be two distinct natural numbers. It is known that $a^2+b|b^2+a$ and that $b^2+a$ is a power of a prime number. Determine the possible values of $a$ and $b$.
2012 IMO Shortlist, N6
Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.
2008 India Regional Mathematical Olympiad, 5
Let $N$ be a ten digit positive integer divisible by $7$. Suppose the first and the last digit of $N$ are interchanged and the resulting number (not necessarily ten digit) is also divisible by $7$ then we say that $N$ is a good integer. How many ten digit good integers are there?
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]
2025 6th Memorial "Aleksandar Blazhevski-Cane", P5
Let $s < t$ be positive integers. Define a sequence by: $a_1 = s, a_2 = t$; $a_3$ is the smallest integer that's greater than $a_2$ and divisible by $a_1$; in general, $a_{n + 1}$ is the smallest integer greater than $a_n$ that's divisible by $a_1, a_2, ..., a_{n - 2}, a_{n - 1}$.
[b]a)[/b] What is the maximum number of odd integers that can appear in such a sequence? (Justify your answer)
[b]b)[/b] Prove that $a_{2025}$ is divisible by $2^{808}$, regardless of the choice of $s$ and $t$.
Proposed by [i]Ilija Jovcevski[/i]
2011 China National Olympiad, 3
Let $m,n$ be positive integer numbers. Prove that there exist infinite many couples of positive integer nubmers $(a,b)$ such that
\[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]
PEN A Problems, 103
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
2021 Baltic Way, 20
Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that
$$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$
2005 IMO Shortlist, 4
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]
2008 Germany Team Selection Test, 2
For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number
\[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}}
\]
but $ 2^{3k \plus{} 1}$ does not.
[i]Author: Waldemar Pompe, Poland[/i]
1962 IMO Shortlist, 1
Find the smallest natural number $n$ which has the following properties:
a) Its decimal representation has a 6 as the last digit.
b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.
2008 Federal Competition For Advanced Students, P1, 1
What is the remainder of the number $1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}$ when divided by $2008$?
2014 India Regional Mathematical Olympiad, 3
Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.
2017 Regional Competition For Advanced Students, 4
Determine all integers $n \geq 2$, satisfying
$$n=a^2+b^2,$$
where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$.
[i]Proposed by Walther Janous[/i]
2020 Turkey MO (2nd round), 1
Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$.
Fractal Edition 1, P1
Is the number $1234567890987654321$ prime?