Found problems: 15460
2017 ITAMO, 6
Prove that there are infinitely many positive integers $m$ such that the number of odd distinct prime factor of $m(m+3)$ is a multiple of $3$.
2023 India Regional Mathematical Olympiad, 1
Let $\mathbb{N}$ be the set of all positive integers and $S=\left\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right\}$. Find the largest positive integer $m$ such that $m$ divides abcd for all $(a, b, c, d) \in S$.
2012 China Second Round Olympiad, 2
Prove that the set $\{2,2^2,\ldots,2^n,\ldots\}$ satisfies the following properties:
[b](1)[/b] For every $a\in A, b\in\mathbb{N}$, if $b<2a-1$, then $b(b+1)$ isn't a multiple of $2a$;
[b](2)[/b] For every positive integer $a\notin A,a\ne 1$, there exists a positive integer $b$, such that $b<2a-1$ and $b(b+1)$ is a multiple of $2a$.
2010 India National Olympiad, 2
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.
2021 JBMO Shortlist, N7
Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$. Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$. Who has a winning strategy?
Proposed by [i]Borche Joshevski, Macedonia[/i]
2017 CMIMC Individual Finals, 1
Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.
2010 Purple Comet Problems, 23
A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
size(150);
defaultpen(linewidth(1));
draw(circle(origin,10)^^circle((3,0),8)^^circle((5,15/4),15/4)^^circle((5,-15/4),15/4));
[/asy]
2019 IMO Shortlist, N7
Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th powers.
[i]Canada[/i]
2012 Dutch Mathematical Olympiad, 3
Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$,
for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.
1988 Poland - Second Round, 4
Prove that for every natural number $ n $, the number $ n^{2n} - n^{n+2} + n^n - 1 $ is divisible by $ (n - 1 )^3 $.
1971 Bulgaria National Olympiad, Problem 1
A natural number is called [i]triangular[/i] if it may be presented in the form $\frac{n(n+1)}2$. Find all values of $a$ $(1\le a\le9)$ for which there exist a triangular number all digit of which are equal to $a$.
2005 Polish MO Finals, 2
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
2023 LMT Fall, 21
Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$
[i]Proposed by Muztaba Syed[/i]
2010 Malaysia National Olympiad, 9
Show that there exist integers $m$ and $n$ such that \[\dfrac{m}{n}=\sqrt[3]{\sqrt{50}+7}-\sqrt[3]{\sqrt{50}-7}.\]
2000 Iran MO (3rd Round), 1
A sequence of natural numbers $c_1, c_2,\dots$ is called [i]perfect[/i] if every natural
number $m$ with $1\le m \le c_1 +\dots+ c_n$ can be represented as
$m =\frac{c_1}{a_1}+\frac{c_2}{a_2}+\dots+\frac{c_n}{a_n}$
Given $n$, find the maximum possible value of $c_n$ in a perfect sequence $(c_i)$.
2005 Denmark MO - Mohr Contest, 4
Fourteen students each write an integer number on the board. When they later meet their math teacher Homer Grog, they tell him that no matter what number they erased on the board, then the remaining numbers could be divided into three groups at once sum. They also tell him that the numbers on the board were integer numbers. Is it now possible for Homer Grog to determine what numbers the students wrote on the board?
2020 AMC 12/AHSME, 21
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$
$\textbf{(A) } 12 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 72$
2013 NIMO Problems, 10
There exist primes $p$ and $q$ such that
\[ pq = 1208925819614629174706176 \times 2^{4404} - 4503599560261633
\times 134217730 \times 2^{2202} + 1. \]
Find the remainder when $p+q$ is divided by $1000$.
[i]Proposed by Evan Chen[/i]
2013 Bosnia and Herzegovina Junior BMO TST, 1
It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers
2014 Stars Of Mathematics, 1
Prove there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^3+y \mid x+y^3$.
([i]Dan Schwarz[/i])
2015 Romania National Olympiad, 2
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $
[b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $
[b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $
2024 Kyiv City MO Round 2, Problem 2
You are given a positive integer $n > 1$. What is the largest possible number of integers that can be chosen from
the set $\{1, 2, 3, \ldots, 2^n\}$ so that for any two different chosen integers $a, b$, the value $a^k + b^k$ is not divisible by $2^n$ for any positive integer $k$?
[i]Proposed by Oleksii Masalitin[/i]
PEN Q Problems, 7
Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
2023 New Zealand MO, 1
For any positive integer $n$ let $n! = 1\times 2\times 3\times ... \times n$. Do there exist infinitely many triples $(p, q, r)$, of positive integers with $p > q > r > 1$ such that the product $p! \cdot q! \cdot r!$$ is a perfect square?
2021 AMC 12/AHSME Fall, 16
Let $a, b,$ and $c$ be positive integers such that $a+b+c=23$ and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^{2}+b^{2}+c^{2}$?
$\textbf{(A)} ~259\qquad\textbf{(B)} ~438\qquad\textbf{(C)} ~516\qquad\textbf{(D)} ~625\qquad\textbf{(E)} ~687$
Proposed by [b]djmathman[/b]