Olympiad problems

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.

2019 Canadian Mathematical Olympiad Qualification, 8

Tags: inequalities , divide , number theory

For $t \ge 2$, define $S(t)$ as the number of times $t$ divides into $t!$. We say that a positive integer $t$ is a [i]peak[/i] if $S(t) > S(u)$ for all values of $u < t$. Prove or disprove the following statement: For every prime $p$, there is an integer $k$ for which $p$ divides $k$ and $k$ is a peak.

2022 China Northern MO, 2

Tags: number theory , divide

(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$, (2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$. Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?

2016 Peru IMO TST, 4

Tags: max , divide , min , positive integer , number theory

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

2018 Saudi Arabia GMO TST, 2

Tags: prime , divisible , divide , number theory

Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.

2020 Denmark MO - Mohr Contest, 3

Tags: number theory , digit , divide

Which positive integers satisfy the following three conditions? a) The number consists of at least two digits. b) The last digit is not zero. c) Inserting a zero between the last two digits yields a number divisible by the original number.

2021 Auckland Mathematical Olympiad, 4

Tags: number theory , power , divide , divisible

Prove that there exist two powers of $7$ whose difference is divisible by $2021$.

2008 Switzerland - Final Round, 6

Tags: odd , divide , divisible , number theory

Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.

1908 Eotvos Mathematical Competition, 1

Tags: divide , number theory , 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 $2^n$.

1986 Tournament Of Towns, (108) 2

Tags: digit , divide , number theory , maximum

A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)

2013 QEDMO 13th or 12th, 2

Tags: number theory , binomial , divide , divisible

Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.

VMEO III 2006 Shortlist, N3

Tags: divide , number theory

Given odd prime $p$. Sequence ${x_n}$ is defined by $x_{n+2}= 4x_{n+1}-x_n$. Choose $x_0,x_1$ such that for every random positive integer $k$, there exists $i\in \mathbb N$ such that $4p^2-8p+1|x_i - (2p)^k$.

2016 Saudi Arabia Pre-TST, 2.4

Tags: divisible , number theory , product , divide

Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that $$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.

2015 Dutch BxMO/EGMO TST, 1

Tags: divisor , divide , number theory

Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

2001 Austria Beginners' Competition, 1

Tags: number theory , divide , divisible

Prove that for every odd positive integer $n$ the number $n^n-n$ is divisible by $24$.

2012 Thailand Mathematical Olympiad, 7

Tags: gcd , divide , number theory , greatest common divisor

Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: coprime , divide , number theory

Let $n$ be a positive integer and $A$ a set containing $8n + 1$ positive integers co-prime with $6$ and less than $30n$. Prove that there exist $a, b \in A$ two different numbers such that $a$ divides $b$.

1949-56 Chisinau City MO, 7

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

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divide

Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.

2014 Czech-Polish-Slovak Junior Match, 4

Tags: divide , digit , number theory , divisible

The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

1989 Tournament Of Towns, (215) 3

Tags: divide , number theory , divisible

Find six distinct positive integers such that the product of any two of them is divisible by their sum. (D. Fomin, Leningrad)

2002 Portugal MO, 4

Tags: number theory , divide , digit

The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.

2011 Saudi Arabia IMO TST, 3

Tags: polynomial , number theory , divide , algebra

Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.

2008 Thailand Mathematical Olympiad, 4

Tags: binomial , divide , number theory , divisible

Let $n$ be a positive integer. Show that $${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.

1996 Estonia National Olympiad, 4

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

Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.