Found problems: 387
2016 Peru IMO TST, 4
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
2000 All-Russian Olympiad Regional Round, 9.2
Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?
2019 Saudi Arabia Pre-TST + Training Tests, 5.1
Let $n$ be a positive integer and $p > n+1$ a prime.
Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$
2003 Chile National Olympiad, 5
Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)
2018 Thailand Mathematical Olympiad, 5
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?
2023 Francophone Mathematical Olympiad, 4
Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.
1997 Singapore MO Open, 2
Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.
2012 Switzerland - Final Round, 7
Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.
2013 Balkan MO Shortlist, A5
Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.
1960 Poland - Second Round, 4
Prove that if $ n $ is a non-negative integer, then number $$
2^{n+2} + 3^{2n+1}$$
is divisible by $7$.
2012 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$
2016 Saudi Arabia BMO TST, 3
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.
2017 Saudi Arabia BMO TST, 1
Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.
VMEO III 2006 Shortlist, N3
Given odd prime $p$. Sequence ${x_n}$ is defined by $x_{n+2}= 4x_{n+1}-x_n$.
Choose $x_0,x_1$ such that for every random positive integer $k$, there exists $i\in \mathbb N$ such that $4p^2-8p+1|x_i - (2p)^k$.
2018 Istmo Centroamericano MO, 1
A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.
1995 Singapore MO Open, 4
Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.
2022 Regional Olympiad of Mexico West, 5
Determine all positive integers $n$ such that $\lfloor \sqrt{n} \rfloor - 1$ divides $n + 1$ and $\lfloor \sqrt{n} \rfloor +2$ divides $ n + 4$.
1986 Austrian-Polish Competition, 7
Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$. Prove that $k$ divides $n!$.
2000 Singapore MO Open, 2
Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.
2016 Saudi Arabia BMO TST, 3
For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.
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 $4!$.
2015 Dutch IMO TST, 5
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
1969 Swedish Mathematical Competition, 5
Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.
2015 Denmark MO - Mohr Contest, 2
The numbers $1, 2, 3, . . . , 624$ are paired in such a way that the sum of the two numbers in each pair is $625$. For example $1$ and $624$ form a pair, and $30$ and $595$ form a pair. In how many of the $312$ pairs does the smaller number evenly divide the larger?
2016 Grand Duchy of Lithuania, 4
Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.