Olympiad problems

1959 Poland - Second Round, 4

Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula $$a_n =3(n^2 + n) + 7$$ Prove that this sequence has the following properties: 1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $, 2( No term of the sequence is the cube of an integer.

1999 Ukraine Team Selection Test, 8

Tags: divide , number theory

Find all pairs $(x,n)$ of positive integers for which $x^n + 2^n + 1$ divides $x^{n+1} +2^{n+1} +1$.

2011 Saudi Arabia Pre-TST, 3.1

Tags: number theory , divide , divisible

Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .

1982 Poland - Second Round, 5

Tags: number theory , divide , divisible

Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.

2005 Chile National Olympiad, 2

Tags: number theory , divide

Let $p$ be a prime number greater than $2$ and let $m, n$ be integers such that: $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}.$$ Prove that $p$ divides $m$.

2009 Chile National Olympiad, 4

Tags: divide , divisible , number theory

Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$

2014 Junior Balkan Team Selection Tests - Moldova, 4

Tags: number theory , divide , divisible

A set $A$ contains $956$ natural numbers between $1$ and $2014$, inclusive. Prove that in the set $A$ there are two numbers $a$ and $b$ such that $a + b$ is divided by $19$.

2017 Puerto Rico Team Selection Test, 3

Tags: number theory , multiple , divide

Given are $n$ integers. Prove that at least one of the following conditions applies: 1) One of the numbers is a multiple of $n$. 2) You can choose $k\le n$ numbers whose sum is a multiple of $ n$.

2016 Czech-Polish-Slovak Junior Match, 3

Tags: combinatorial geometry , combinatorics , divide , 3d geometry , prism , geometry

Find all integers $n \ge 3$ with the following property: it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$. Poland

2005 Greece JBMO TST, 4

Tags: number theory , divide , factorial

Find all the positive integers $n , n\ge 3$ such that $n\mid (n-2)!$

1985 Czech And Slovak Olympiad IIIA, 6

Tags: number theory , divide

Prove that for every natural number $n > 1$ there exists a suquence $a_1$,$a_2$, $...$, $a_n$ of the numbers $1,2,...,n$ such that for each $k \in \{1,2,...,n-1\}$ the number $a_{k+1}$ divides $a_1+a_2+...+a_k$.

2012 Abels Math Contest (Norwegian MO) Final, 3b

Tags: number theory , divisible , divide

Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?

2003 Switzerland Team Selection Test, 4

Tags: divide , max , number theory

Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.

2009 Thailand Mathematical Olympiad, 10

Tags: number theory , divide , divisible

Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.

2015 Saudi Arabia GMO TST, 4

Tags: number theory , divide , divisible

Let $p$ be an odd prime number. Prove that there exists a unique integer $k$ such that $0 \le k \le p^2$ and $p^2$ divides $k(k + 1)(k + 2) ... (k + p - 3) - 1$. Malik Talbi

1997 Chile National Olympiad, 2

Tags: number theory , divisible , divide

Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$. Prove that $a \equiv n \equiv 1 \mod 10$ .

2020 Colombia National Olympiad, 4

Tags: arithmetic sequence , divide , number theory

Find all of the sequences $a_1, a_2, a_3, . . .$ of real numbers that satisfy the following property: given any sequence $b_1, b_2, b_3, . . .$ of positive integers such that for all $n \ge 1$ we have $b_n \ne b_{n+1}$ and $b_n | b_{n+1}$, then the sub-sequence $a_{b_1}, a_{b_2}, a_{b_3}, . . .$ is an arithmetic progression.

2003 Cuba MO, 2

Tags: number theory , divide

Prove that if $$\frac{p}{q}=1-\frac{1}{2} + \frac{1}{3}- \frac{1}{4} + ... -\frac{1}{1334} + \frac{1}{1335}$$ where $p, q \in Z_+$ then $p$ is divisible by $2003$.

2023 Greece Junior Math Olympiad, 4

Tags: number theory , divide , divisor

Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.

2023 Durer Math Competition Finals, 15

Tags: number theory , divide

What is the biggest positive integer which divides $p^4 - q^4$ for all primes $p$ and $q$ greater than $10$?

2015 Balkan MO Shortlist, N2

Tags: recurrence relation , number theory with sequences , sequence , divide , number theory , algebra

Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$, and $ a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$. Prove that $n^2$ divides $a_n$ for infinite $n$. (Romania)

1958 Kurschak Competition, 2

Tags: number theory , divide , divisible

Show that if $m$ and $n$ are integers such that $m^2 + mn + n^2$ is divisible by $9$, then they must both be divisible by $3$.

2015 Singapore Junior Math Olympiad, 1

Tags: multiple , number theory , divide , digit

Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

2020 Malaysia IMONST 2, 3

Tags: number theory , divisible , divide

Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.

2011 Indonesia TST, 4

Tags: number theory , prime , sum of divisors , divide

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.