Found problems: 545
2005 India IMO Training Camp, 2
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2024 Middle European Mathematical Olympiad, 8
Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that
\[a_ia_{i+1} \mid k-a_i^2\]
for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all
integers $n \ge M$.
2010 Belarus Team Selection Test, 3.2
Prove that there exists a positive integer $n$ such that $n^6 + 31n^4 - 900\vdots 2009 \cdot 2010 \cdot 2011$.
(I. Losev, I. Voronovich)
2014 District Olympiad, 4
Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with
the properties:
[list=a]
[*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $
[*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]
2016 Turkey Team Selection Test, 5
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .
1983 IMO Shortlist, 5
Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.
2007 Alexandru Myller, 1
Solve $ x^3-y^3=2xy+7 $ in integers.
2013 Ukraine Team Selection Test, 11
Specified natural number $a$. Prove that there are an infinite number of prime numbers $p$ such that for some natural $n$ the number $2^{2^n} + a$ is divisible by $p$.
1991 IMO Shortlist, 13
Given any integer $ n \geq 2,$ assume that the integers $ a_1, a_2, \ldots, a_n$ are not divisible by $ n$ and, moreover, that $ n$ does not divide $ \sum^n_{i\equal{}1} a_i.$ Prove that there exist at least $ n$ different sequences $ (e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $ \sum^n_{i\equal{}1} e_i \cdot a_i$ is divisible by $ n.$
2005 China Team Selection Test, 1
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
1999 Mongolian Mathematical Olympiad, Problem 1
Prove that for any positive integer $k$ there exist infinitely many positive integers $m$ such that $3^k\mid m^3+10$.
2021 Taiwan TST Round 1, N
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
1993 Mexico National Olympiad, 6
$p$ is an odd prime. Show that $p$ divides $n(n+1)(n+2)(n+3) + 1$ for some integer $n$ iff $p$ divides $m^2 - 5$ for some integer $m$.
2018 Thailand TSTST, 3
Find all pairs of integers $m, n \geq 2$ such that $$n\mid 1+m^{3^n}+m^{2\cdot 3^n}.$$
2021 Science ON all problems, 1
Find all sequences of positive integers $(a_n)_{n\ge 1}$ which satisfy
$$a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1)$$
for all $n\in \mathbb{Z}_{\ge 1}$.
[i](Bogdan Blaga)[/i]
2018 Regional Olympiad of Mexico Southeast, 2
Let $n=\frac{2^{2018}-1}{3}$. Prove that $n$ divides $2^n-2$.
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.
2020 Israel National Olympiad, 4
At the start of the day, the four numbers $(a_0,b_0,c_0,d_0)$ were written on the board. Every minute, Danny replaces the four numbers written on the board with new ones according to the following rule: if the numbers written on the board are $(a,b,c,d)$, then Danny first calculates the numbers
\begin{align*}
a'&=a+4b+16c+64d\\
b'&=b+4c+16d+64a\\
c'&=c+4d+16a+64b\\
d'&=d+4a+16b+64c
\end{align*}
and replaces the numbers $(a,b,c,d)$ with the numbers $(a'd',c'd',c'b',b'a')$.
For which initial quadruples $(a_0,b_0,c_0,d_0)$, will Danny write at some point a quadruple of numbers all of which are divisible by $5780^{5780}$?
2020 Switzerland - Final Round, 1
Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to \mathbb N$ such that for every $m,n\in \mathbb N$, \[
f(m)+f(n)\mid m+n.
\]
1966 IMO Shortlist, 42
Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by
$n.$
[b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$
1969 IMO Longlists, 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 $\frac{\phi(d)}{b}?$
2000 Belarus Team Selection Test, 8.2
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.
1969 IMO Shortlist, 34
$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$
2018 OMMock - Mexico National Olympiad Mock Exam, 4
For each positive integer $n$ let $s(n)$ denote the sum of the decimal digits of $n$. Find all pairs of positive integers $(a, b)$ with $a > b$ which simultaneously satisfy the following two conditions
$$a \mid b + s(a)$$
$$b \mid a + s(b)$$
[i]Proposed by Victor DomÃnguez[/i]
1998 Nordic, 3
(a) For which positive numbers $n$ does there exist a sequence $x_1, x_2, ..., x_n$, which contains each of the numbers $1, 2, ..., n$ exactly once and for which $x_1 + x_2 +... + x_k$ is divisible by $k$ for each $k = 1, 2,...., n$?
(b) Does there exist an infinite sequence $x_1, x_2, x_3, ..., $ which contains every positive integer exactly once and such that $x_1 + x_2 +... + x_k$ is divisible by $k$ for every positive integer $k$?