Olympiad problems

2021 Francophone Mathematical Olympiad, 1

Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.

2006 Thailand Mathematical Olympiad, 14

Tags: number theory , divide

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

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 $19$.

1911 Eotvos Mathematical Competition, 3

Tags: number theory , divide , divisible

Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.

2014 Peru MO (ONEM), 3

Tags: number theory , divide

a) Let $a, b, c$ be positive integers such that $ab + b + 1$, $bc + c + 1$ and $ca + a + 1$ are divisors of the number $abc - 1$, prove that $a = b = c$. b) Find all triples $(a, b, c)$ of positive integers such that the product $$(ab - b + 1)(bc - c + 1)(ca - a + 1)$$ is a divisor of the number $(abc + 1)^2$.

2006 May Olympiad, 1

Tags: number theory , divide

Determine all pairs of natural numbers $a$ and $b$ such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ they are natural numbers.

1999 Abels Math Contest (Norwegian MO), 2b

Tags: number theory , divide , divisible

If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$

2015 Puerto Rico Team Selection Test, 8

Tags: divisible , number theory , multiple , divide

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

2016 Saudi Arabia BMO TST, 3

Tags: divide , divisible , number theory

For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.

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

2011 QEDMO 10th, 3

Tags: number theory , multiple , divide

Let $a, b$ be positive integers such that $a^2 + ab + 1$ a multiple of $b^2 + ab + 1$. Prove that $a = b$.

1959 Poland - Second Round, 4

Tags: perfect cube , number theory , divide

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.

1996 Austrian-Polish Competition, 1

Tags: number theory , divide , divisible , digit

Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits

2015 Costa Rica - Final Round, N3

Tags: number theory , divide

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

2000 Tournament Of Towns, 5

Tags: number theory , divisible , consecutive , divide

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)

2015 Thailand Mathematical Olympiad, 1

Tags: number theory , divide , recurrence relation , divisible

Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.

2012 China Northern MO, 8

Tags: number theory , divide

Assume $p$ is a prime number. If there is a positive integer $a$ such that $p!|(a^p + 1)$, prove that : (1) $(a+1, \frac{a^p+1}{a+1}) = p$ (2) $\frac{a^p+1}{a+1}$ has no prime factors less than $p$. (3) $p!|(a +1) $.

2014 India PRMO, 13

Tags: number theory , integer , divide

For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?

2000 Portugal MO, 3

Tags: divisor , number theory , divide

Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

2023 Grand Duchy of Lithuania, 4

Tags: number theory , divide , factorial

Note that $k\ge 1$ for an odd natural number $$k! ! = k \cdot (k - 2) \cdot ... \cdot 1.$$ Prove that $2^n$ divides $(2^n -1)!! -1$ for all $n \ge 3$.

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 .

2021 Czech and Slovak Olympiad III A, 4

Tags: number theory , divisor , divide

Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$, where $d (0) = d (1) = 0$ and for $k> 1$, $ d (k)$ is the [i]superdivisor [/i] of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$). (Tomáš Bárta)

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 $150$.

2012 Brazil Team Selection Test, 2

Tags: number theory , divide , divisible

Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.

2011 Argentina National Olympiad, 5

Tags: number theory , divide , divisible

Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.