Found problems: 408
1987 Tournament Of Towns, (159) 3
Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .
2016 Saudi Arabia IMO TST, 3
Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that:
$\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot 2016$.
$\bullet$ The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$.
Find the maximum value of $n$
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 $2005$?
2000 Czech And Slovak Olympiad IIIA, 1
Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.
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
2011 IMAR Test, 4
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
2017 Junior Regional Olympiad - FBH, 4
Let $n$ and $k$ be positive integers for which we have $4$ statements:
$i)$ $n+1$ is divisible with $k$
$ii)$ $n=2k+5$
$iii)$ $n+k$ is divisible with $3$
$iv)$ $n+7k$ is prime
Determine all possible values for $n$ and $k$, if out of the $4$ statements, three of them are true and one is false
2022 New Zealand MO, 6
Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.
2013 Costa Rica - Final Round, 6
Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.
2008 Dutch Mathematical Olympiad, 3
Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included).
Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.
2021 Saudi Arabia Training Tests, 39
Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.
1989 Tournament Of Towns, (205) 3
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$?
(M . I. Gusarov)
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 $p$.
1999 Abels Math Contest (Norwegian MO), 2b
If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$
2018 Czech-Polish-Slovak Junior Match, 4
Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.
1952 Moscow Mathematical Olympiad, 226
Seven chips are numbered $1, 2, 3, 4, 5, 6, 7$. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.
2019 Durer Math Competition Finals, 5
We want to write down as many distinct positive integers as possible, so that no two numbers on our list have a sum or a difference divisible by $2019$. At most how many integers can appear on such a list?
1974 Dutch Mathematical Olympiad, 2
$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$
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 $p^2$
VMEO I 2004, 2
The Fibonacci numbers $(F_n)_{n=1}^{\infty}$ are defined as follows:
$$F_1 = F_2 = 1, F_n = F_{n-2} + F_{n-1}, n = 3, 4, ...$$
Assume $p$ is a prime greater than $3$. With $m$ being a natural number greater than $3$, find all $n$ numbers such that $F_n$ is divisible by $p^m$.
1954 Moscow Mathematical Olympiad, 267
Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$
then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.
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
2012 Austria Beginners' Competition, 1
Let $a, b, c$ and $d$ be four integers such that $7a + 8b = 14c + 28d$.
Prove that the product $a\cdot b$ is always divisible by $14$.
2006 Estonia National Olympiad, 2
Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares.
a) Give an example of such numbers $a, b$ and $c$.
b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$
2007 Bosnia and Herzegovina Junior BMO TST, 2
Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$.
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