Olympiad problems

2020 Regional Olympiad of Mexico Northeast, 4

Let $n > 1$ be an integer and $p$ be a prime. Prove that if $n|p-1$ and $p|n^3-1$, then $4p-3$ is a perfect square.

2011 Belarus Team Selection Test, 2

Tags: number theory , divides , divisible

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

2019 New Zealand MO, 4

Tags: number theory , divides , divisible , multiple

Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.

2008 Switzerland - Final Round, 6

Tags: odd , number theory , divides , divisible

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

2017 Auckland Mathematical Olympiad, 3

Tags: number theory , divisible , Digits , divides

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 $13$.

1994 Tournament Of Towns, (417) 5

Tags: number theory , Digits , divisible , divides

Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

2020 Silk Road, 1

Tags: numberr theory , inequalities , divisible , divides

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 $.

2010 Saudi Arabia IMO TST, 3

Tags: recurrence relation , divides , number theory

Consider the sequence $a_1 = 3$ and $a_{n + 1} =\frac{3a_n^2+1}{2}-a_n$ for $n = 1 ,2 ,...$. Prove that if $n$ is a power of $3$ then $n$ divides $a_n$ .

2022 Austrian MO National Competition, 1

Tags: function , number theory , divides

Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divides , divided

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 ^ {*} $.

1960 Poland - Second Round, 4

Tags: number theory , divides , divisible

Prove that if $ n $ is a non-negative integer, then number $$ 2^{n+2} + 3^{2n+1}$$ is divisible by $7$.

2003 Estonia National Olympiad, 4

Tags: number theory , infinite , Integer , floor function , divisible , divides

Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.

1958 Polish MO Finals, 1

Tags: number theory , perfect cube , divides

Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .

2016 Saudi Arabia GMO TST, 3

Tags: divides , divisor , algebra , polynomial

Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions: $x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$

1972 Spain Mathematical Olympiad, 7

Tags: number theory , multiple , divides , divisible

Prove that for every positive integer $n$, the number $$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$ is a multiple of $8$.

2007 Thailand Mathematical Olympiad, 11

Tags: number theory , combinatorics , divisible , divides

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

2017 Saudi Arabia JBMO TST, 2

Tags: number theory , divides

A positive integer $k > 1$ is called nice if for any pair $(m, n)$ of positive integers satisfying the condition $kn + m | km + n$ we have $n | m$. 1. Prove that $5$ is a nice number. 2. Find all the nice numbers.

2016 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , divides , Digits

Find the largest integer $d$ divides all three numbers $abc, bca$ and $cab$ with $a, b$ and $c$ being some nonzero and mutually different digits. Czech Republic

2010 Thailand Mathematical Olympiad, 3

Tags: number theory , divisible , divides

Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.

1989 Tournament Of Towns, (210) 4

Tags: number theory , divides , divisible

Prove that if $k$ is an even positive integer then it is possible to write the integers from $1$ to $k-1$ in such an order that the sum of no set of successive numbers is divisible by $k$ .

2014 Chile National Olympiad, 4

Tags: number theory , divides , divisible

Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$.

2023 Greece Junior Math Olympiad, 4

Tags: number theory , Divisors , divides

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 $5b-3$.

2018 Saudi Arabia GMO TST, 2

Tags: number theory , divides , divisible , prime

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

2016 Flanders Math Olympiad, 2

Tags: Power , number theory , divides , divisor

Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.

2015 Costa Rica - Final Round, N3

Tags: number theory , divides

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