Olympiad problems

2015 Saudi Arabia Pre-TST, 3.3

Let $(a_n)_{n\ge0}$ be a sequence of positive integers such that $a^2_n$ divides $a_{n-1}a_{n+1}$, for all $n \ge 1$. Prove that if there exists an integer $k \ge 2$ such that $a_k$ and $a_1$ are relatively prime, then $a_1$ divides $a_0$. (Malik Talbi)

2019 Durer Math Competition Finals, 11

Tags: sum , divide , divisible , number theory

What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?

2017 Junior Balkan Team Selection Tests - Romania, 3

Tags: divide , number theory

Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent: a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold, b) $n$ divides $504$.

2017 Gulf Math Olympiad, 4

Tags: ceiling function , power of 2 , divide , maximum , number theory

1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ . 2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

2011 Tournament of Towns, 6

Tags: divide , divisor , number theory , power of 2

Prove that the integer $1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}$ is a multiple of $2^n$ but not a multiple of $2^{n+1}$.

2008 Postal Coaching, 3

Tags: number theory , product , divide , divisible

Prove that there exists an innite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

2000 Chile National Olympiad, 5

Tags: power of 2 , divide , divisible , number theory

Let $n$ be a positive number. Prove that there exists an integer $N =\overline{m_1m_2...m_n}$ with $m_i \in \{1, 2\}$ which is divisible by $2^n$.

2006 QEDMO 2nd, 5

Tags: number theory , divide

For any natural number $m$, we denote by $\phi (m)$ the number of integers $k$ relatively prime to $m$ and satisfying $1 \le k \le m$. Determine all positive integers $n$ such that for every integer $k > n^2$, we have $n | \phi (nk + 1)$. (Daniel Harrer)

1994 Austrian-Polish Competition, 7

Tags: number theory , divide , divisible

Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.

1980 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , divide , divisible

Prove that for every nonnegative integer $ k$ there is a product $$(k + 1)(k + 2)...(k + 1980)$$ divisible by $ 1980^{197}$.

2016 Saudi Arabia Pre-TST, 1.3

Tags: number theory , odd , divide

Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.

1908 Eotvos Mathematical Competition, 1

Tags: number theory , divide , divisible

Given two odd integers $a$ and $b$; prove that $a^3 -b^3$ is divisible by $2^n$ if and only if $a-b$ is divisible by $2^n$.

1965 German National Olympiad, 2

Tags: number theory , divide

Determine which of the prime numbers $2,3,5,7,11,13,109,151,491$ divide $z =1963^{1965} -1963$.

2009 China Northern MO, 3

Tags: number theory , divide

Given $26$ different positive integers , in any six numbers of the $26$ integers , there are at least two numbers , one can be devided by another. Then prove : There exists six numbers , one of them can be devided by the other five numbers .

2001 Kazakhstan National Olympiad, 1

Tags: divide , divisible , number theory

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

2006 Thailand Mathematical Olympiad, 14

Tags: number theory , divide

Find the smallest positive integer $n$ such that $2549 | n^{2545} - 2$.

2019 Costa Rica - Final Round, 5

Tags: number theory , divide

We have an a sequence such that $a_n = 2 \cdot 10^{n + 1} + 19$. Determine all the primes $p$, with $p \le 19$, for which there exists some $n \ge 1$ such that $p$ divides $a_n$.

2012 Tournament of Towns, 3

Tags: number theory , divide , polynomial , divisible

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 $2n - 1$.

2022 Chile Junior Math Olympiad, 4

Tags: number theory , divide

Let $S$ be the sum of all products $ab$ where $a$ and $b$ are distinct elements of the set $\{1,2,...,46\}$. Prove that $47$ divides $S$.

2015 Saudi Arabia GMO TST, 4

Tags: number theory , divide , divisible

Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$. Malik Talbi

1965 Dutch Mathematical Olympiad, 2

Tags: number theory , perfect square , divide , divisible

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

2014 Saudi Arabia Pre-TST, 4.1

Tags: number theory , inequalities , divide , divisible

Let $p$ be a prime number and $n \ge 2$ a positive integer, such that $p | (n^6 -1)$. Prove that $n > \sqrt{p}-1$.

2010 Saudi Arabia IMO TST, 1

Tags: number theory , divide , divisible

Find all pairs $(m,n)$ of integers, $m ,n \ge 2$ such that $mn - 1$ divides $n^3 - 1$.

2017 QEDMO 15th, 10

Tags: number theory , divisor , divide

Let $p> 3$ be a prime number and let $q = \frac{4^p-1}{3}$. Show that $q$ is a composite integer as well is a divisor of $2^{q-1}- 1$.

2017 Regional Olympiad of Mexico West, 6

Tags: divide , number theory

A [i]change [/i] in a natural number $n$ consists of adding a pair of zeros between two digits or at the end of the decimal representation of $n$. A [i]countryman [/i] of $n$ is a number that can be obtained from one or more changes in $n$. For example. $40041$, $4410000$ and $4004001$ are all countrymen from $441$. Determine all the natural numbers $n$ for which there is a natural number m with the property that $n$ divides $m$ and all the countrymen of $m$.