Found problems: 367
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 $3$.
2003 Junior Balkan Team Selection Tests - Romania, 3
A set of $2003$ positive integers is given. Show that one can find two elements such that their sum is not a divisor of the sum of the other elements.
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 Saudi Arabia Pre-TST, 1.2
Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.
2003 Junior Balkan Team Selection Tests - Romania, 2
Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.
2009 Kyiv Mathematical Festival, 5
The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?
2017 Saudi Arabia Pre-TST + Training Tests, 1
Let $m, n, k$ and $l$ be positive integers with $n \ne 1$ such that $n^k + mn^l + 1$ divides $n^{k+l }- 1$.
Prove that either $m = 1$ and $l = 2k$, or $l | k$ and $m =\frac{n^{k-l} - 1}{n^l - 1}$.
2013 Cuba MO, 8
Prove that there are infinitely many pairs $(a, b)$ of positive integers with the following properties:
$\bullet$ $a+b$ divides $ab+1$,
$\bullet$ $a-b$ divides $ab -1$,
$\bullet$ $b > 2$ and $a > b\sqrt3 - 1$.
2012 Brazil Team Selection Test, 4
Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$
2018 District Olympiad, 2
Find the pairs of integers $(a, b)$ such that $a^2 + 2b^2 + 2a +1$ is a divisor of $2ab$.
2023 Peru MO (ONEM), 1
We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$.
a) What is the smallest positive integer that divides exactly two elements of $M$?
b) What is the largest positive integer that divides exactly two elements of $M$?
2020 Denmark MO - Mohr Contest, 3
Which positive integers satisfy the following three conditions?
a) The number consists of at least two digits.
b) The last digit is not zero.
c) Inserting a zero between the last two digits yields a number divisible by the original number.
2017 Latvia Baltic Way TST, 16
Strings $a_1, a_2, ... , a_{2016}$ and $b_1, b_2, ... , b_{2016}$ each contain all natural numbers from $1$ to $2016$ exactly once each (in other words, they are both permutations of the numbers $1, 2, ..., 2016$). Prove that different indices $i$ and $j$ can be found such that $a_ib_i- a_jb_j$ is divisible by $2017$.
1970 Poland - Second Round, 3
Prove the theorem:
There is no natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $.
2012 India Regional Mathematical Olympiad, 2
Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.
1991 Bundeswettbewerb Mathematik, 2
Let $g$ be an even positive integer and $f(n) = g^n + 1$ , $(n \in N^* )$.
Prove that for every positive integer $n$ we have:
a) $f(n)$ divides each of the numbers $f(3n), f(5n), f(7n)$
b) $f(n)$ is relative prime to each of the numbers $f(2n), f(4n),f(6n),...$
1999 Singapore MO Open, 2
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer
1993 Swedish Mathematical Competition, 1
An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.
2009 Thailand Mathematical Olympiad, 10
Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.
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 $4!$.
1911 Eotvos Mathematical Competition, 3
Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.
2019 Final Mathematical Cup, 3
Determine every prime numbers $p$ and $q , p \le q$ for which $pq | (5^p - 2^ p )(7^q -2 ^q )$
2008 Postal Coaching, 2
Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$.
[Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.]
[hide=Hint]$n$ is squarefree[/hide]
2007 Switzerland - Final Round, 9
Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.
2017 Junior Balkan Team Selection Tests - Romania, 3
Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent:
a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold,
b) $n$ divides $504$.