Found problems: 15460
2020 Grand Duchy of Lithuania, 4
We shall call an integer n [i]cute [/i] if it can be written in the form $n = a^2 + b^3 + c^3 + d^5$,
where $a, b, c$ and $d$ are integers.
a) Determine if the number $2020$ is cute.
b) Find all cute integers
2008 Bosnia Herzegovina Team Selection Test, 2
Find all pairs of positive integers $ m$ and $ n$ that satisfy (both) following conditions:
(i) $ m^{2}\minus{}n$ divides $ m\plus{}n^{2}$
(ii) $ n^{2}\minus{}m$ divides $ n\plus{}m^{2}$
2005 Paraguay Mathematical Olympiad, 3
The complete list of the three-digit palindrome numbers is written in ascending order: $$101, 111, 121, 131,... , 979, 989, 999.$$ Then eight consecutive palindrome numbers are eliminated and the numbers that remain in the list are added, obtaining $46.150$. Determine the eight erased palindrome numbers .
1984 IMO Longlists, 30
Decide whether it is possible to color the $1984$ natural numbers $1, 2, 3, \cdots, 1984$ using $15$ colors so that no geometric sequence of length $3$ of the same color exists.
2016 Ukraine Team Selection Test, 12
Suppose that $a_0, a_1, \cdots $ and $b_0, b_1, \cdots$ are two sequences of positive integers such that $a_0, b_0 \ge 2$ and \[ a_{n+1} = \gcd{(a_n, b_n)} + 1, \qquad b_{n+1} = \operatorname{lcm}{(a_n, b_n)} - 1. \] Show that the sequence $a_n$ is eventually periodic; in other words, there exist integers $N \ge 0$ and $t > 0$ such that $a_{n+t} = a_n$ for all $n \ge N$.
2021 Durer Math Competition Finals, 8
Benedek wrote the following $300 $ statements on a piece of paper.
$2 | 1!$
$3 | 1! \,\,\, 3 | 2!$
$4 | 1! \,\,\, 4 | 2! \,\,\, 4 | 3!$
$5 | 1! \,\,\, 5 | 2! \,\,\, 5 | 3! \,\,\, 5 | 4!$
$...$
$24 | 1! \,\,\, 24 | 2! \,\,\, 24 | 3! \,\,\, 24 | 4! \,\,\, · · · \,\,\, 24 | 23!$
$25 | 1! \,\,\, 25 | 2! \,\,\, 25 | 3! \,\,\, 25 | 4! \,\,\, · · · \,\,\, 25 | 23! \,\,\, 25 | 24!$
How many true statements did Benedek write down?
The symbol | denotes divisibility, e.g. $6 | 4!$ means that $6$ is a divisor of number $4!$.
2007 China Team Selection Test, 1
Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \plus{} b \minus{} 1$ is a power of prime number $ ; a^2 \plus{} b \plus{} 1$ can divide $ b^2 \minus{} a^3 \minus{} 1,$ but it can't divide $ (a \plus{} b \minus{} 1)^2.$
2024 Princeton University Math Competition, A3 / B5
Let $\sigma$ be a permutation of the set $S := \{1, 2, \ldots , 100\},$ such that $\sigma(a+b) \equiv \sigma(a)+\sigma(b) \pmod{100}$ whenever $a, b, a + b \in S.$ Denote by $f(s)$ the sum of the distinct values $\sigma(s)$ can take over all possible $\sigma$s satisfying the given condition. What is the nonnegative difference between the maximum and minimum value $f$ takes on when ranging over all $s \in S$?
1990 IMO Longlists, 84
Let $n \geq 4$ be an integer. $a_1, a_2, \ldots, a_n \in (0, 2n)$ are $n$ distinct integers. Prove that there exists a subset of the set $\{a_1, a_2, \ldots, a_n \}$ such that the sum of its elements is divisible by $2n.$
2002 China Team Selection Test, 3
For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that
\[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\
b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\
c^2 &= \alpha\beta\gamma. \end{cases} \]
Also, let $ \lambda$ be a real number that satisfies the condition
\[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\]
Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.
2022 Azerbaijan JBMO TST, N1
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to
factorials of some positive integers.
Proposed by [i]Nikola Velov, Macedonia[/i]
2023 Brazil Team Selection Test, 4
Find all positive integers $n$ with the following property: There are only a finite number of positive multiples of $n$ that have exactly $n$ positive divisors.
2023 4th Memorial "Aleksandar Blazhevski-Cane", P6
Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
[b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$.
[b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$.
[i]Proposed by Nikola Velov[/i]
1989 Mexico National Olympiad, 4
Find the smallest possible natural number $n = \overline{a_m ...a_2a_1a_0} $ (in decimal system) such that the number $r = \overline{a_1a_0a_m ..._20} $ equals $2n$.
VMEO IV 2015, 12.2
Given a positive integer $k$. Prove that there are infinitely many positive integers $n$ satisfy the following conditions at the same time:
a) $n$ has at least $k$ distinct prime divisors
b) All prime divisors other than $3$ of $n$ have the form $4t+1$, with $t$ some positive integer.
c) $n | 2^{\sigma(n)}-1$
Here $\sigma(n)$ demotes the sum of the positive integer divisors of $n$.
2022 Saudi Arabia IMO TST, 3
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2018 Indonesia MO, 1
Let $a$ be a positive integer such that $\gcd(an+1, 2n+1) = 1$ for all integer $n$.
a) Prove that $\gcd(a-2, 2n+1) = 1$ for all integer $n$.
b) Find all possible $a$.
2011 Thailand Mathematical Olympiad, 5
Find all $n$ such that \[n = d (n) ^ 4\]
Where $d (n)$ is the number of divisors of $n$, for example $n = 2 \cdot 3\cdot 5\implies d (n) = 2 \cdot 2\cdot 2$.
2022 SG Originals, Q3
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$.
[i]Proposed by DVDthe1st[/i]
2020 HMNT (HMMO), 5
The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the room number as shown on the sign. The smallest interval containing all possible values of $\frac{M}{N}$ can be expressed as $[\frac{a}{b}, \frac{c}{d} )$ where $a,b,c,d$ are positive integers with $\gcd(a,b) = \gcd(c,d) = 1$. Compute $1000a+100b+10c+d$.
2005 South africa National Olympiad, 1
Five numbers are chosen from the diagram below, such that no two numbers are chosen from the same row or from the same column. Prove that their sum is always the same.
\[\begin{array}{|c|c|c|c|c|}\hline
1&4&7&10&13\\ \hline
16&19&22&25&28\\ \hline
31&34&37&40&43\\ \hline
46&49&52&55&58\\ \hline
61&64&67&70&73\\ \hline
\end{array}\]
2002 Greece National Olympiad, 4
(a) Positive integers $p,q,r,a$ satisfy $pq=ra^2$, where $r$ is prime and $p,q$ are relatively prime. Prove that one of the numbers $p,q$ is a perfect square.
(b) Examine if there exists a prime $p$ such that $p(2^{p+1}-1)$ is a perfect square.
2013 NIMO Problems, 8
The number $\frac{1}{2}$ is written on a blackboard. For a real number $c$ with $0 < c < 1$, a [i]$c$-splay[/i] is an operation in which every number $x$ on the board is erased and replaced by the two numbers $cx$ and $1-c(1-x)$. A [i]splay-sequence[/i] $C = (c_1,c_2,c_3,c_4)$ is an application of a $c_i$-splay for $i=1,2,3,4$ in that order, and its [i]power[/i] is defined by $P(C) = c_1c_2c_3c_4$.
Let $S$ be the set of splay-sequences which yield the numbers $\frac{1}{17}, \frac{2}{17}, \dots, \frac{16}{17}$ on the blackboard in some order. If $\sum_{C \in S} P(C) = \tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Lewis Chen[/i]
2012 CHMMC Spring, 1
Let $a, b, c$ be positive integers. Suppose that $(a + b)(a + c) = 77$ and $(a + b)(b + c) = 56$. Find $(a + c)(b + c)$.
2017 Harvard-MIT Mathematics Tournament, 7
Let $p$ be a prime. A [i]complete residue class modulo $p$[/i] is a set containing at least one element equivalent to $k \pmod{p}$ for all $k$.
(a) Show that there exists an $n$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.
(b) Show that there exists an $n \le p^2$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.