Olympiad problems

2008 Estonia Team Selection Test, 4

Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.

2014 Regional Olympiad of Mexico Center Zone, 1

Tags: number theory , divisible , divide , product

Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

2022 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , divisible

The number $2022$ is written on the board. In each step, we replace one of the $2$ digits with the number $2022$. For example $$2022 \Rightarrow 2020222 \Rightarrow 2020220222 \Rightarrow ...$$ After how many steps can a number divisible by $22$ be written on the board? Specify all options.

2013 Saudi Arabia Pre-TST, 2.1

Tags: number theory , remainder , divide , divisible

Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.

2012 QEDMO 11th, 12

Tags: number theory , divisor , divide , divisible

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.

2001 Kazakhstan National Olympiad, 1

Tags: divisible , divide , number theory

Prove that there are infinitely many natural numbers $ n $ such that $ 2 ^ n + 3 ^ n $ is divisible by $ n $.

2019 Hanoi Open Mathematics Competitions, 6

Tags: number theory , divisible

What is the largest positive integer $n$ such that $10 \times 11 \times 12 \times ... \times 50$ is divisible by $10^n$?

1953 Moscow Mathematical Olympiad, 236

Tags: divisible , number theory

Prove that $n^2 + 8n + 15$ is not divisible by $n + 4$ for any positive integer $n$.

1977 Dutch Mathematical Olympiad, 3

Tags: divisible , number theory , divide , 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 $7$.

VMEO III 2006 Shortlist, N4

Tags: divisible , digit , divide , number theory

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 $n$.

2008 Dutch Mathematical Olympiad, 3

Tags: number theory , divisible , divide , combinatorics

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 $8$.

2018 Puerto Rico Team Selection Test, 3

Tags: number theory , divide , divisible

Let $A$ be a set of $m$ positive integers where $m\ge 1$. Show that there exists a nonempty subset $B$ of $A$ such that the sum of all the elements of $B$ is divisible by $m$.

2012 NZMOC Camp Selection Problems, 2

Tags: number theory , divisible , consecutive , divide , sum

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

2015 Hanoi Open Mathematics Competitions, 3

Tags: algebra , number theory , divisible

The sum of all even positive integers less than $100$ those are not divisible by $3$ is (A): $938$, (B): $940$, (C): $1634$, (D): $1638$, (E): None of the above.

2005 iTest, 25

Tags: number theory , divide , divisible

Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?

1977 Swedish Mathematical Competition, 1

Tags: divide , divisible , number theory

$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.

2019 New Zealand MO, 1

Tags: divisible , number theory

How many positive integers less than $2019$ are divisible by either $18$ or $21$, but not both?

2000 Czech And Slovak Olympiad IIIA, 1

Tags: divisible , number theory , even , divide

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.

2013 Balkan MO Shortlist, N4

Tags: divisible , number theory , sum of powers , divisor

Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$.

2008 Estonia Team Selection Test, 4

Tags: number theory , sequence , divisible , consecutive

Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.

2017 Latvia Baltic Way TST, 16

Tags: divisible , combinatorics , divide , number theory

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 $2017$.

1987 Tournament Of Towns, (159) 3

Tags: divisible , divide , number theory

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 $a$ .

1972 All Soviet Union Mathematical Olympiad, 162

Tags: relatively prime , divisible , number theory

a) Let $a,n,m$ be natural numbers, $a > 1$. Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$. b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime. Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$.

1986 Tournament Of Towns, (129) 4

Tags: number theory , divisible , factorial , divisor , divide

We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

2016 Grand Duchy of Lithuania, 4

Tags: divisible , number theory , divide

Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.