Found problems: 545
2022 European Mathematical Cup, 2
We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$.
a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?
2022 Bundeswettbewerb Mathematik, 1
Five squirrels together have a supply of 2022 nuts. On the first day 2 nuts are added, on the second day 4 nuts, on the third day 6 nuts and so on, i.e. on each further day 2 nuts more are added than on the day before.
At the end of any day the squirrels divide the stock among themselves. Is it possible that they all receive the same number of nuts and that no nut is left over?
1997 Romania Team Selection Test, 2
Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$.
[i]Marcel Tena[/i]
2017 China Northern MO, 2
Prove that there exist infinitely many integers \(n\) which satisfy \(2017^2 | 1^n + 2^n + ... + 2017^n\).
1992 IMO Longlists, 63
Let $a$ and $b$ be integers. Prove that $\frac{2a^2-1}{b^2+2}$ is not an integer.
2010 IMO Shortlist, 4
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$
[list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good.
[*][b](b)[/b] Show that all 2010-good pairs are very good.[/list]
[i]Proposed by Okan Tekman, Turkey[/i]
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]
2004 Unirea, 2
Find the arithmetic sequences of $ 5 $ integers $ n_1,n_2,n_3,n_4,n_5 $ that verify $ 5|n_1,2|n_2,11|n_3,7|n_4,17|n_5. $
2009 Ukraine Team Selection Test, 7
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
2012 IMO Shortlist, N8
Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.
2011 Morocco TST, 1
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
1987 IMO Shortlist, 8
(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$
(b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$
[i]Proposed by Hungary.[/i]
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.
2021 Science ON all problems, 2
Find all pairs $(p,q)$ of prime numbers such that
$$p^q-4~|~q^p-1.$$
[i](Vlad Robu)[/i]
2021 Peru Iberoamerican Team Selection Test, P1
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 $$n=\frac{ab+3b+8}{a^2+b+3}.$$
2008 Germany Team Selection Test, 1
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$.
[i]Author: Stephan Wagner, Austria[/i]
2022 Taiwan TST Round 1, 2
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 $$n=\frac{ab+3b+8}{a^2+b+3}.$$
1998 IMO, 4
Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.
2021 Thailand TST, 1
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 IMO Shortlist, 4
Show that for any finite set $S$ of distinct positive integers, we can find a set $T \supseteq S$ such that every member of $T$ divides the sum of all the members of $T$.
[b]Original Statement:[/b]
A finite set of (distinct) positive integers is called a [b]DS-set[/b] if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some [b]DS-set[/b].
1969 IMO Longlists, 13
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?
2017 Germany Team Selection Test, 3
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]
2017 Thailand TSTST, 2
$\text{(i)}$ Does there exist a positive integer $m > 2016^{2016}$ such that $\frac{2016^m-m^{2016}}{m+2016}$ is a positive integer?
$\text{(ii)}$ Does there exist a positive integer $m > 2017^{2017}$ such that $\frac{2017^m-m^{2017}}{m+2017}$ is a positive integer?
[i](Serbia MO 2016 P1)[/i]
2021 Balkan MO Shortlist, C2
Let $K$ and $N > K$ be fixed positive integers. Let $n$ be a positive integer and let $a_1, a_2, ..., a_n$ be distinct integers. Suppose that whenever $m_1, m_2, ..., m_n$ are integers, not all equal to $0$, such that $\mid{m_i}\mid \le K$ for each $i$, then the sum
$$\sum_{i = 1}^{n} m_ia_i$$
is not divisible by $N$. What is the largest possible value of $n$?
[i]Proposed by Ilija Jovcevski, North Macedonia[/i]
2020 USA TSTST, 4
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]
[i]Yang Liu[/i]