Found problems: 408
1993 All-Russian Olympiad Regional Round, 11.2
Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.
2017 Czech-Polish-Slovak Junior Match, 3
How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?
2022 APMO, 1
Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.
2004 All-Russian Olympiad Regional Round, 9.5
The cells of a $100 \times 100$ table contain non-zero numbers. It turned out that all $100$ hundred-digit numbers written horizontally are divisible by 11. Could it be that exactly $99$ hundred-digit numbers written vertically are also divisible by $11$?
2003 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$.
(H. Nestra)
2015 Saudi Arabia JBMO TST, 1
A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$
2007 BAMO, 4
Let $N$ be the number of ordered pairs $(x,y)$ of integers such that $x^2+xy+y^2 \le 2007$.
Remember, integers may be positive, negative, or zero!
(a) Prove that $N$ is odd.
(b) Prove that $N$ is not divisible by $3$.
2015 JBMO Shortlist, NT3
a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$
b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$
PS. a) original from Albania
b) modified by problem selecting committee
1991 Chile National Olympiad, 4
Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.
2015 Caucasus Mathematical Olympiad, 1
Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?
2012 Brazil Team Selection Test, 2
Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.
1949-56 Chisinau City MO, 9
Prove that for any integer $n$ the number $n (n^2 + 5)$ is divisible by $6$.
2015 Puerto Rico Team Selection Test, 3
Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.
1999 Switzerland Team Selection Test, 6
Prove that if $m$ and $n$ are positive integers such that $m^2 + n^2 - m$ is divisible by $2mn$, then $m$ is a perfect square.
2021 Bosnia and Herzegovina Team Selection Test, 2
Let $p > 2$ be a prime number. Prove that there is a permutation $k_1, k_2, ..., k_{p-1}$ of numbers $1,2,...,p-1$ such that the number $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$ is divisible by $p$.
Note: The numbers $k_1, k_2, ..., k_{p-1}$ are a permutation of the numbers $1,2,...,p-1$ if each of of numbers $1,2,...,p-1$ appears exactly once among the numbers $k_1, k_2, ..., k_{p-1}$.
2012 QEDMO 11th, 12
Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.
2017 May Olympiad, 5
We will say that two positive integers $a$ and $b$ form a [i]suitable pair[/i] if $a+b$ divides $ab$ (its sum divides its multiplication). Find $24$ positive integers that can be distribute into $12$ suitable pairs, and so that each integer number appears in only one pair and the largest of the $24$ numbers is as small as possible.
2012 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$
2020 Swedish Mathematical Competition, 1
How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?
2018 Saudi Arabia GMO TST, 2
Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.
2016 Saudi Arabia Pre-TST, 1.4
Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$.
1. Show that $F(r)$ is a positive integer for any prime $r \ne p$.
2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$.
3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.
2000 Tournament Of Towns, 2
Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$.
(A Spivak)
2009 Hanoi Open Mathematics Competitions, 1
Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$
Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$
VMEO IV 2015, 11.3
How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?
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)