Found problems: 536
2017 Balkan MO Shortlist, N4
Find all pairs of positive integers $(x,y)$ , such that $x^2$ is divisible by $2xy^2 -y^3 +1$.
1989 Irish Math Olympiad, 5
Let $x = a_1a_2 \dots a_n$ be an n-digit number, where $a_1, a_2, \dots , an (a_1 \neq 0)$ are the digits. The $n$ numbers $ x_1 = x = a_1 a_2 ... a_n, $ $ x_2 = a_n a_1 ... a_{n-1}, $ $ x_3 = a_{n-1} a_n a _1 ... a_{n-2} $ ,
$ x_4 = a_{n-2} a_{n-1} a_n a_1 , ... a_{n-3} , $ $ ... , x_n = a_2 a_3 ... a_n a_1$
are said to be obtained from $x$ by the cyclic permutation of digits. [For example, if $n = 5$ and $x = 37001$, then the numbers are $x_1 = 37001, x_2 = 13700, $ $x_3 = 01370(= 1370), x_4 = 00137(= 137), $ $ x_5 = 70013.]$
Find, with proof, (i) the smallest natural number n for which there exists an n-digit number x such that the n numbers obtained from x by the cyclic permutation of digits are all divisible by 1989; and (ii) the smallest natural number x with this property.
1984 IMO, 2
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
2016 Ukraine Team Selection Test, 7
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
1992 IMO Longlists, 63
Let $a$ and $b$ be integers. Prove that $\frac{2a^2-1}{b^2+2}$ is not an integer.
2009 IMO Shortlist, 3
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
2021 IMO Shortlist, N1
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$
2022 Vietnam National Olympiad, 4
For every pair of positive integers $(n,m)$ with $n<m$, denote $s(n,m)$ be the number of positive integers such that the number is in the range $[n,m]$ and the number is coprime with $m$. Find all positive integers $m\ge 2$ such that $m$ satisfy these condition:
i) $\frac{s(n,m)}{m-n} \ge \frac{s(1,m)}{m}$ for all $n=1,2,...,m-1$;
ii) $2022^m+1$ is divisible by $m^2$
1974 IMO Longlists, 30
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
1996 Estonia Team Selection Test, 1
Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.
2017 China Northern MO, 2
Prove that there exist infinitely many integers \(n\) which satisfy \(2017^2 | 1^n + 2^n + ... + 2017^n\).
2023 Polish MO Finals, 1
Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.
1969 IMO Shortlist, 23
$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$
2020 Romania EGMO TST, P1
Determine if for any positive integers $a,b,c$ there exist pairwise distinct non-negative integers $A,B,C$ which are greater than $2019$ such that $a+A,b+B$ and $c+C$ divide $ABC$.
2018 China Team Selection Test, 6
Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions:
(1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$
(2) For all $n\ge M$, $f(n)\ge 0$
(3) For all $n>m>M$, $n-m|f(n)-f(m)$
Show that $a$ is a perfect $r$-th power.
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 $k$.
2010 Peru IMO TST, 9
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
1984 IMO Shortlist, 12
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
2023 Romanian Master of Mathematics Shortlist, N1
Let $n$ be a positive integer. Let $S$ be a set of ordered pairs $(x, y)$ such that $1\leq x \leq n$ and $0 \leq y \leq n$ in each pair, and there are no pairs $(a, b)$ and $(c, d)$ of different elements in $S$ such that $a^2+b^2$ divides both
$ac+bd$ and $ad - bc$. In terms of $n$, determine the size of the largest possible set $S$.
2020 Tournament Of Towns, 1
$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?
A. Gribalko
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.
2019 Switzerland - Final Round, 8
An integer $n\ge2$ is called [i]resistant[/i], if it is coprime to the sum of all its divisors (including $1$ and $n$).
Determine the maximum number of consecutive resistant numbers.
For instance:
* $n=5$ has sum of divisors $S=6$ and hence is resistant.
* $n=6$ has sum of divisors $S=12$ and hence is not resistant.
* $n=8$ has sum of divisors $S=15$ and hence is resistant.
* $n=18$ has sum of divisors $S=39$ and hence is not resistant.
2017 Bosnia Herzegovina Team Selection Test, 2
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
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 $b$.
2018 Pan-African Shortlist, N4
Let $S$ be a set of $49$-digit numbers $n$, with the property that each of the digits $1, 2, 3, \dots, 7$ appears in the decimal expansion of $n$ seven times (and $8, 9$ and $0$ do not appear). Show that no two distinct elements of $S$ divide each other.