Found problems: 15460
1949 Miklós Schweitzer, 3
Let $ p$ be an odd prime number and $ a_1,a_2,...,a_p$ and $ b_1,b_2,...,b_p$ two arbitrary permutations of the numbers $ 1,2,...,p$ . Show that the least positive residues modulo $ p$ of the numbers $ a_1b_1, a_2b_2,...,a_pb_p$ never form a permutation of the numbers $ 1,2,...,p$.
IV Soros Olympiad 1997 - 98 (Russia), 11.2
Find the three-digit number that has the greatest number of different divisors.
2023 Azerbaijan National Mathematical Olympiad, 1
For any natural number, let's call the numbers formed from its digits and have the same "digit" arrangement with the initial number as the "partial numbers". For example, the partial numbers of $149$ are ${1, 4, 9, 14,19, 49, 149},$ and the partial numbers of $313$ are ${3, 1, 31,33, 13, 313}.$ Find all natural numbers whose partial numbers are all prime. Justify your opinion.
1972 IMO Longlists, 31
Find values of $n\in \mathbb{N}$ for which the fraction $\frac{3^n-2}{2^n-3}$ is reducible.
2021 Junior Balkan Team Selection Tests - Romania, P2
Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.
1997 Bosnia and Herzegovina Team Selection Test, 6
Let $k$, $m$ and $n$ be integers such that $1<n \leq m-1 \leq k$. Find maximum size of subset $S$ of set $\{1,2,...,k\}$ such that sum of any $n$ different elements from $S$ is not:
$a)$ equal to $m$,
$b)$ exceeding $m$
2024 Ecuador NMO (OMEC), 2
Let $s(n)$ the sum of digits of $n$. Find the greatest 3-digits number $m$ such that $3s(m)=s(3m)$.
2006 ITAMO, 2
Solve $p^n+144=m^2$ where $m,n\in \mathbb{N}$ and $p$ is a prime number.
2009 Baltic Way, 8
Determine all positive integers $n$ for which there exists a partition of the set
\[\{n,n+1,n+2,\ldots ,n+8\}\]
into two subsets such that the product of all elements of the first subset is equal to the product of all elements of the second subset.
2017 BMT Spring, 11
Ben picks a positive number $n$ less than $2017$ uniformly at random. Then Rex, starting with the number $ 1$, repeatedly multiplies his number by $n$ and then finds the remainder when dividing by $2017$. Rex does this until he gets back to the number $ 1$. What is the probability that, during this process, Rex reaches every positive number less than $2017$ before returning back to $ 1$?
1977 IMO Longlists, 12
Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$
2016 Benelux, 2
Let $n$ be a positive integer. Suppose that its positive divisors can be partitioned into pairs (i.e. can be split in groups of two) in such a way that the sum of each pair is a prime number. Prove that these prime numbers are distinct and that none of these are a divisor of $n.$
1961 All-Soviet Union Olympiad, 4
Given are arbitrary integers $a,b,p$. Prove that there always exist relatively prime integers $k$ and $\ell$ such that $ak+b\ell$ is divisible by $p$.
2008 May Olympiad, 1
How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?
2004 German National Olympiad, 3
Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.
2016 Balkan MO, 3
Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer.
[i]Note: A monic polynomial has a leading coefficient equal to 1.[/i]
[i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]
2020 Italy National Olympiad, #2
Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied:
1- $b>a$ and $b-a$ is a prime number
2- The last digit of the number $a+b$ is $3$
3- The number $ab$ is a square of an integer.
2021 Romanian Master of Mathematics Shortlist, N2
We call a set of positive integers [i]suitable [/i] if none of its elements is coprime to the sum of all
elements of that set. Given a real number $\varepsilon \in (0,1)$, prove that, for all large enough positive
integers $N$, there exists a suitable set of size at least $\varepsilon N$, each element of which is at most $N$.
2016 Kosovo National Mathematical Olympiad, 1
Find all couples $(m,n)$ of positive integers such that satisfied $m^2+1=n^2+2016$ .
2012 China Team Selection Test, 2
For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if $\tau (m) < \tau (n)$ for all $m < n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.
1998 Croatia National Olympiad, Problem 4
Among any $79$ consecutive natural numbers there exists one whose sum of digits is divisible by $13$. Find a sequence of $78$ consecutive natural numbers for which the above statement fails.
2011 NZMOC Camp Selection Problems, 1
Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$
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2013 Spain Mathematical Olympiad, 5
Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions
$i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct).
$ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$
2010 Germany Team Selection Test, 3
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
2003 Moldova Team Selection Test, 1
Each side of an arbitrarly triangle is divided into $ 2002$ congruent segments. After that, each vertex is joined with all "division" points on the opposite side.
Prove that the number of the regions formed, in which the triangle is divided, is divisible by $ 6$.
[i]Proposer[/i]: [b]Dorian Croitoru[/b]