Found problems: 15460
2021 New Zealand MO, 3
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$?
2011 Kazakhstan National Olympiad, 3
Given are the odd integers $m> 1$, $k$, and a prime $p$ such that $p> mk +1$. Prove that $p^{2}\mid {\binom{k}{k}}^{m}+{\binom{k+1}{k}}^{m}+\cdots+{\binom{p-1}{k}}^{m}$.
1991 IMO Shortlist, 18
Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.\]
2013 Canadian Mathematical Olympiad Qualification Repechage, 3
A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.
2017 CMIMC Number Theory, 6
Find the largest positive integer $N$ satisfying the following properties:
[list]
[*]$N$ is divisible by $7$;
[*]Swapping the $i^{\text{th}}$ and $j^{\text{th}}$ digits of $N$ (for any $i$ and $j$ with $i\neq j$) gives an integer which is $\textit{not}$ divisible by $7$.
[/list]
2002 Croatia National Olympiad, Problem 4
Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.
2011 IMO Shortlist, 8
Let $k \in \mathbb{Z}^+$ and set $n=2^k+1.$ Prove that $n$ is a prime number if and only if the following holds: there is a permutation $a_{1},\ldots,a_{n-1}$ of the numbers $1,2, \ldots, n-1$ and a sequence of integers $g_{1},\ldots,g_{n-1},$ such that $n$ divides $g^{a_i}_i - a_{i+1}$ for every $i \in \{1,2,\ldots,n-1\},$ where we set $a_n = a_1.$
[i]Proposed by Vasily Astakhov, Russia[/i]
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P3
Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$.
[i]Proposed by Viktor Simjanoski[/i]
1999 Akdeniz University MO, 1
Let $n$'s positive divisors sum is $T(n)$. For all $n \geq 3$'s prove that,
$$(T(n))^3<n^4$$
2020 Benelux, 4
A divisor $d$ of a positive integer $n$ is said to be a [i]close[/i] divisor of $n$ if $\sqrt{n}<d<2\sqrt{n}$. Does there exist a positive integer with exactly $2020$ close divisors?
2017 India IMO Training Camp, 3
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
2020 BMT Fall, 3
Compute the remainder when $98!$ is divided by $101$.
2024 Germany Team Selection Test, 2
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.
[i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]
2023 Durer Math Competition (First Round), 5
Let $n \ge 3$ be an integer. Timi thought of $n$ different real numbers and then wrote down the numbers which she could produce as the product of two different numbers she had in mind. At most how many different positive prime numbers did she write down (depending on $n$)?
1980 Czech And Slovak Olympiad IIIA, 1
Prove that for every nonnegative integer $ k$ there is a product
$$(k + 1)(k + 2)...(k + 1980)$$
divisible by $ 1980^{197}$.
1998 Slovenia National Olympiad, Problem 1
Let $n$ be a positive integer. If the number $1998$ is written in base $n$, a three-digit number with the sum of digits equal to $24$ is obtained. Find all possible values of $n$.
2016 Bosnia And Herzegovina - Regional Olympiad, 4
Let $a$ and $b$ be distinct positive integers, bigger that $10^6$, such that $(a+b)^3$ is divisible with $ab$. Prove that $ \mid a-b \mid > 10^4$
2022 Belarusian National Olympiad, 10.1
Prove that for any positive integer one can place all it's divisor on a circle such that among any two neighbours one is a multiple of the other
JOM 2025, 2
Let $n$ be a positive integer. Navinim writes down all positive square numbers that divide $n$ on a blackboard. For each number $k$ on the blackboard, Navagem replaces it with $d(k)$. Show that the sum of all numbers on the blackboard now is a perfect square. (Note: $d(k)$ denotes the number of divisors of $k$.)
[i](Proposed by Ivan Chan Guan Yu)[/i]
2006 Thailand Mathematical Olympiad, 13
Compute the remainder when $\underbrace{\hbox{11...1}}_{\hbox{1862}}$ is divided by $2006$
1984 IMO Shortlist, 16
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 $a=1$.
2017 Hanoi Open Mathematics Competitions, 3
Suppose $n^2 + 4n + 25$ is a perfect square. How many such non-negative integers $n$'s are there?
(A): $1$ (B): $2$ (C): $4$ (D): $6$ (E): None of the above.
2008 Princeton University Math Competition, A7
Find the smallest positive integer $n$ such that $32^n = 167x + 2$ for some integer $x$
2000 Harvard-MIT Mathematics Tournament, 40
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ and relatively prime to $n$. Find all natural numbers $n$ and primes $p$ such that $\phi(n)=\phi(np)$.
2022 Ecuador NMO (OMEC), 1
Prove that it is impossible to divide a square with side length $7$ into exactly $36$ squares with integer side lengths.