Found problems: 367
2003 Singapore Senior Math Olympiad, 1
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.
2000 Kazakhstan National Olympiad, 5
Let the number $ p $ be a prime divisor of the number $ 2 ^ {2 ^ k} + 1 $. Prove that $ p-1 $ is divisible by $ 2 ^ {k + 1} $.
2005 Thailand Mathematical Olympiad, 12
Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.
1951 Polish MO Finals, 2
What digits should be placed instead of zeros in the third and fifth places in the number $3000003$ to obtain a number divisible by $13$?
2013 Danube Mathematical Competition, 2
Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.
2012 NZMOC Camp Selection Problems, 4
A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.
2004 Thailand Mathematical Olympiad, 15
Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$.
2005 All-Russian Olympiad Regional Round, 10.5
Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?
2013 Thailand Mathematical Olympiad, 1
Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$
2019 Durer Math Competition Finals, 6
Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?
2018 Bosnia and Herzegovina EGMO TST, 2
Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds
$$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$
2002 Junior Balkan Team Selection Tests - Romania, 1
Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers.
Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.
2013 Saudi Arabia Pre-TST, 2.1
Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.
2009 District Olympiad, 1
Let $m$ and $n$ be positive integers such that $5$ divides $2^n + 3^m$. Prove that $5$ divides $2^m + 3^n$.
2015 Gulf Math Olympiad, 1
a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$.
b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$.
c) Suppose that $p$ is an odd prime, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$.
d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer.
Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.
VMEO IV 2015, 10.3
Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$
2021 Francophone Mathematical Olympiad, 1
Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.
2015 Switzerland - Final Round, 9
Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties:
1) $1 \le ai \le p$ for all $i = 1, . . . , p$.
2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$.
3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.
2018 Grand Duchy of Lithuania, 4
Find all positive integers $n$ for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.
2012 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$
1949-56 Chisinau City MO, 7
Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.
2021 Regional Competition For Advanced Students, 4
Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$.
(Walther Janous)
2002 Estonia National Olympiad, 1
The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.
2011 Saudi Arabia IMO TST, 3
Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.
2007 IMAC Arhimede, 4
Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$