Found problems: 15460
2021 Brazil National Olympiad, 5
Find all triples of non-negative integers \((a, b, c)\) such that \[a^{2}+b^{2}+c^{2} = a b c+1.\]
2024 Assara - South Russian Girl's MO, 7
Find all positive integers $n$ for such the following condition holds:
"If $a$, $b$ and $c$ are positive integers such are all numbers \[ a^2+2ab+b^2,\ b^2+2bc+c^2, \ c^2+2ca+a^2 \] are divisible by $n$, then $(a+b+c)^2$ is also divisible by $n$."
[i]G.M.Sharafetdinova[/i]
1991 Tournament Of Towns, (291) 1
Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied:
$$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$
2006 MOP Homework, 6
Find all integers $n$ for which there exists an equiangular $n$-gon whose side lengths are distinct rational numbers.
1993 Irish Math Olympiad, 2
A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1\plus{}a_2\plus{}...\plus{}a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.
2021 Saudi Arabia JBMO TST, 4
Let us call a set of positive integers nice if the number of its elements equals to the average of its numbers. Call a positive integer $n$ an [i]amazing[/i] number if the set $\{1, 2 , . . . , n\}$ can be partitioned into nice subsets.
a) Prove that every perfect square is amazing.
b) Show that there are infinitely many positive integers which are not amazing.
2010 Postal Coaching, 5
Prove that there exist a set of $2010$ natural numbers such that product of any $1006 $ numbers is divisible by product of remaining $1004$ numbers.
2024 Chile Classification NMO Juniors, 2
Find all pairs of positive integers \((a, b)\) such that
\[
\frac{a+1}{b} , \frac{b+1}{a}
\]
are both positive integers.
1978 IMO Longlists, 10
Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.
2014 IMC, 4
Let $n>6$ be a perfect number, and let $n=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k}$ be its prime factorisation with $1<p_1<\dots <p_k$. Prove that $e_1$ is an even number.
A number $n$ is [i]perfect[/i] if $s(n)=2n$, where $s(n)$ is the sum of the divisors of $n$.
(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)
2022 Thailand Mathematical Olympiad, 2
Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that
$$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$
for every positive integers $m,n$. Determine the minimum possible value of
$$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$
across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.
2018 Costa Rica - Final Round, N3
Let $a$ and $b$ be positive integers such that $2a^2 + a = 3b^2 + b$. Prove that $a-b$ is a perfect square.
2022 Czech-Polish-Slovak Junior Match, 4
Find all triples $(a, b, c)$ of integers that satisfy the equations
$ a + b = c$ and $a^2 + b^3 = c^2$
2012 IFYM, Sozopol, 2
Let $p$ and $q=4p+1$ be prime numbers. Determine the least power $i$ of 2 for which $2^i\equiv 1\,(mod\, q)$.
2020 Bundeswettbewerb Mathematik, 1
Show that there are infinitely many perfect squares of the form $50^m-50^n$, but no perfect square of the form $2020^m+2020^n$, where $m$ and $n$ are positive integers.
2018 Iran Team Selection Test, 3
Let $a_1,a_2,a_3,\cdots $ be an infinite sequence of distinct integers. Prove that there are infinitely many primes $p$ that distinct positive integers $i,j,k$ can be found such that $p\mid a_ia_ja_k-1$.
[i]Proposed by Mohsen Jamali[/i]
1998 India Regional Mathematical Olympiad, 2
Let $n$ be a positive integer and $p_1, p_2, p_3, \ldots p_n$ be $n$ prime numbers all larger than $5$ such that $6$ divides $p_1 ^2 + p_2 ^2 + p_3 ^2 + \cdots p_n ^2$. prove that $6$ divides $n$.
2019 PUMaC Geometry A, 3
Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?
1996 Bosnia and Herzegovina Team Selection Test, 6
Let $a$ and $b$ be two integers which are coprime and let $n$ be one variable integer. Determine probability that number of solutions $(x,y)$, where $x$ and $y$ are nonnegative integers, of equation $ax+by=n$ is $\left\lfloor \frac{n}{ab} \right\rfloor + 1$
2022 Cono Sur, 1
A positive integer is [i]happy[/i] if:
1. All its digits are different and not $0$,
2. One of its digits is equal to the sum of the other digits.
For example, 253 is a [i]happy[/i] number. How many [i]happy[/i] numbers are there?
MBMT Team Rounds, 2020.43
Let $\sigma_k(n)$ be the sum of the $k^{th}$ powers of the divisors of $n$. For all $k \ge 2$ and all $n \ge 3$, we have that $$\frac{\sigma_k(n)}{n^{k+2}} (2020n + 2019)^2 > m.$$ Find the largest possible value of $m$.
2003 South africa National Olympiad, 5
Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.
2011 May Olympiad, 5
We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?
2008 Junior Balkan Team Selection Tests - Romania, 3
Solve in prime numbers $ 2p^q \minus{} q^p \equal{} 7$.
2008 Dutch IMO TST, 4
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.
Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.