Olympiad problems

1997 All-Russian Olympiad Regional Round, 10.3

Natural numbers $m$ and $n$ are given. Prove that the number $2^n-1$ is divisible by the number $(2^m -1)^2$ if and only if the number $n$ is divisible by the number $m(2^m-1)$.

2020 Silk Road, 1

Tags: number theory , inequalities , divisible , divide

Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.

2013 Saudi Arabia BMO TST, 4

Tags: divide , divisible , number theory

Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.

2020 Denmark MO - Mohr Contest, 3

Tags: digit , number theory , 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.

1989 Tournament Of Towns, (215) 3

Tags: number theory , divide , divisible

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

2009 District Olympiad, 1

Tags: number theory , divide

Let $m$ and $n$ be positive integers such that $5$ divides $2^n + 3^m$. Prove that $5$ divides $2^m + 3^n$.

1979 Chisinau City MO, 174

Tags: number theory , odd , divide , divisible

Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.

2018 Costa Rica - Final Round, 5

Tags: number theory , divisible , divide

Let $a$ and $ b$ be even numbers, such that $M = (a + b)^2-ab$ is a multiple of $5$. Consider the following statements: I) The unit digits of $a^3$ and $b^3$ are different. II) $M$ is divisible by $100$. Please indicate which of the above statements are true with certainty.

2018 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , divide , divisible

For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$. Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$

1963 German National Olympiad, 1

Tags: number theory , remainder , divide

a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number! b) Does this also apply when dividing a prime number by $60$? Justify your answer!

2017 Auckland Mathematical Olympiad, 3

Tags: number theory , divisible , digit , divide

The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.

1978 Chisinau City MO, 159

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

Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.

2013 Saudi Arabia Pre-TST, 1.2

Tags: number theory , divide , divisible

Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.

2008 Postal Coaching, 2

Tags: number theory , prime , prime number , divide

Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

1993 Swedish Mathematical Competition, 1

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

An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.

2024 Austrian MO Regional Competition, 4

Tags: number theory , divisible , divide

Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

2007 Thailand Mathematical Olympiad, 11

Tags: number theory , combinatorics , divisible , divide

Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$.

2007 Cuba MO, 5

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

Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.

2011 Belarus Team Selection Test, 2

Tags: number theory , divide , divisible

Do they exist natural numbers $m,x,y$ such that $$m^2 +25 \vdots (2011^x-1007^y) ?$$ S. Finskii

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 $n \ge 3$.

2003 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , divisor , sum , divide , combinatorics

A set of $2003$ positive integers is given. Show that one can find two elements such that their sum is not a divisor of the sum of the other elements.

1979 Czech And Slovak Olympiad IIIA, 6

Tags: number theory , divide , prime

Find all natural numbers $n$, $n < 10^7$, for which: If natural number $m$, $1 < m < n$, is not divisible by $n$, then $m$ is prime.

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 $A$?

2017 Saudi Arabia IMO TST, 3

Tags: number theory , divide , divisible

Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.

2013 Thailand Mathematical Olympiad, 5

Tags: sum of digits , number theory , digit , divide , divisible

Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.