Olympiad problems

2007 Estonia National Olympiad, 4

Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third. a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient. b) Give an example of one of these larger numbers $a, b$ and $c$

2003 Singapore Senior Math Olympiad, 1

Tags: perfect square , divisible , divide , number theory

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.

1997 Tournament Of Towns, (537) 2

Tags: number theory , divisible , divide

Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$. (BR Frenkin)

2015 Saudi Arabia BMO TST, 4

Tags: divide , divisible , number theory

Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$. Lê Anh Vinh

2017 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , divide

Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that $m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.

2019 Paraguay Mathematical Olympiad, 4

Tags: number theory , divisible , divide

Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.

2015 Costa Rica - Final Round, N3

Tags: number theory , divide

Find all the pairs $a,b \in N$ such that $ab-1 |a^2 + 1$.

2022 Junior Balkan Team Selection Tests - Moldova, 11

Tags: divide , number theory

Find all ordered pairs of positive integers $(m, n)$ such that $2m$ divides the number $3n - 2$, and $2n$ divides the number $3m - 2$.

1976 Spain Mathematical Olympiad, 4

Tags: number theory , divide

Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.

2024 Regional Competition For Advanced Students, 4

Tags: number theory , divisible , divide

Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

1947 Moscow Mathematical Olympiad, 136

Tags: 13-gon , parallelogram , convex , divide , combinatorics

Prove that no convex $13$-gon can be cut into parallelograms.

1977 Dutch Mathematical Olympiad, 3

Tags: number theory , divide , divisible , multiple

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 Latvia Baltic Way TST, 16

Tags: combinatorics , number theory , divide , divisible

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 $2017$.

2004 Denmark MO - Mohr Contest, 2

Tags: number theory , divide , divisible

Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

1999 Tournament Of Towns, 3

Tags: number theory , divide , divisible , sum of squares

Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

2007 Switzerland - Final Round, 2

Tags: number theory , divide , divisible

Let $a, b, c$ be three integers such that $a + b + c$ is divisible by $13$. Prove that $$a^{2007}+b^{2007}+c^{2007}+2 \cdot 2007abc$$ is divisible by $13$.

1952 Moscow Mathematical Olympiad, 232

Tags: polynomial , algebra , divide

Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.

1975 Swedish Mathematical Competition, 5

Tags: number theory , divide

Show that $n$ divides $2^n + 1$ for infinitely many positive integers $n$.

1998 Tournament Of Towns, 3

Tags: combinatorics , divisible , divide

Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.) (G Galperin)

2000 Tournament Of Towns, 5

Tags: consecutive , number theory , divide , divisible

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)

2018 Stanford Mathematics Tournament, 1

Tags: number theory , divide , divisible

Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.

2015 Saudi Arabia IMO TST, 1

Tags: number theory , divide , divisible , gcd

Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$. Trần Nam Dũng

2023 Assara - South Russian Girl's MO, 2

Tags: number theory , divisible , divide

The natural numbers $a$ and $b$ are such that $a^a$ is divisible by $b^b$. Can we say that then $a$ is divisible by $b$?

1998 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , divide

Let $X$ be a finite set of positive integers. Prove that for every subset $A$ of $X$, there is a subset $B$ of $X$, with the following property: For each element $ e$ of $X$, $e$ divides an odd number of elements of $B$, if and only if $e$ is an element of $A$.

2021 Saudi Arabia JBMO TST, 3

Tags: number theory , divisible , divide

We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.