Found problems: 408
VMEO IV 2015, 11.3
How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?
1992 All Soviet Union Mathematical Olympiad, 562
Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?
2016 Peru IMO TST, 16
Find all pairs $ (m, n)$ of positive integers that have the following property:
For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.
2016 Ecuador NMO (OMEC), 6
A positive integer $n$ is "[i]olympic[/i]" if there are $n$ non-negative integers $x_1, x_2, ..., x_n$ that satisfy that:
$\bullet$ There is at least one positive integer $j$: $1 \le j \le n$ such that $x_j \ne 0$.
$\bullet$ For any way of choosing $n$ numbers $c_1, c_2, ..., c_n$ from the set $\{-1, 0, 1\}$, where not all $c_i$ are equal to zero, it is true that the sum $c_1x_1 + c_2x_2 +... + c_nx_n$ is not divisible by $n^3$.
Find the largest positive "olympic" integer.
1980 Tournament Of Towns, (003) 3
If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.
2020 Kosovo National Mathematical Olympiad, 2
Let $a_1,a_2,...,a_n$ be integers such that $a_1^{20}+a_2^{20}+...+a_n^{20}$ is divisible by $2020$. Show that $a_1^{2020}+a_2^{2020}+...+a_n^{2020}$ is divisible by $2020$.
1989 Tournament Of Towns, (217) 1
Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers.
( M . N . Gusarov, Leningrad)
2011 Argentina National Olympiad, 5
Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.
2012 Tournament of Towns, 3
Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.
2011 Belarus Team Selection Test, 1
Let $g(n)$ be the number of all $n$-digit natural numbers each consisting only of digits $0,1,2,3$ (but not nessesarily all of them) such that the sum of no two neighbouring digits equals $2$. Determine whether $g(2010)$ and $g(2011)$ are divisible by $11$.
I.Kozlov
1977 Dutch Mathematical Olympiad, 3
From each set $ \{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.
2017 Switzerland - Final Round, 4
Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold:
$$pq | n^p + 2$$
$$n + 2 | n^p + q^p.$$
Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.
2018 India PRMO, 3
Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.
VMEO III 2006 Shortlist, N4
Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.
2000 Tournament Of Towns, 5
What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ?
(S Tokarev)
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 Abels Math Contest (Norwegian MO) Final, 3b
Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?
2017 Hanoi Open Mathematics Competitions, 7
Let two positive integers $x, y$ satisfy the condition $44 /( x^2 + y^2)$.
Determine the smallest value of $T = x^3 + y^3$.
1981 Swedish Mathematical Competition, 3
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.
2022 New Zealand MO, 5
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.
2004 Thailand Mathematical Olympiad, 10
Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.
2021 Polish Junior MO Second Round, 3
Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.
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: $1,2,3,...$)
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 $2005$?
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 $k_1, k_2, ..., k_{p-1}$.