Olympiad problems

2005 Thailand Mathematical Olympiad, 2

Let $S $ be a set of three distinct integers. Show that there are $a, b \in S$ such that $a \ne b$ and $10 | a^3b - ab^3$.

1949 Moscow Mathematical Olympiad, 156

Tags: number theory , divisible

Prove that $27 195^8 - 10 887^8 + 10 152^8$ is divisible by $26 460$.

2021 Dutch IMO TST, 3

Tags: divide , divisible , number theory

Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.

2008 Tournament Of Towns, 4

Tags: number theory , factorial , divisible

Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

2013 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divide , divisible

Find all pairs of integers $(x,y)$ satisfying the following condition: [i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ [/i] Tournament of Towns

1993 All-Russian Olympiad Regional Round, 9.2

Tags: number theory , divide , divisible

Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.

1980 All Soviet Union Mathematical Olympiad, 284

Tags: number theory , divisible

All the two-digit numbers from $19$ to $80$ are written in a line without spaces. Is the obtained number $192021....7980$ divisible by $1980$?

2019 Durer Math Competition Finals, 11

Tags: sum , divide , divisible , number theory

What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?

2003 Estonia Team Selection Test, 2

Tags: number theory , divisible

Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$. (H. Nestra)

1981 Swedish Mathematical Competition, 3

Tags: polynomial , divisible , algebra

Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.

1982 Polish MO Finals, 5

Tags: divisible , sequence , sum

Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.

2018 Tuymaada Olympiad, 5

Tags: number theory , divisible , prime number

A prime $p$ and a positive integer $n$ are given. The product $$(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1)$$ is divisible by $p^3$. Prove that $p \leq n+1$. [i]Proposed by Z. Luria[/i]

2008 Postal Coaching, 3

Tags: number theory , product , divide , divisible

Prove that there exists an innite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

2000 Chile National Olympiad, 5

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

Let $n$ be a positive number. Prove that there exists an integer $N =\overline{m_1m_2...m_n}$ with $m_i \in \{1, 2\}$ which is divisible by $2^n$.

1999 Israel Grosman Mathematical Olympiad, 1

Tags: divisible , number theory

For any $16$ positive integers $n,a_1,a_2,...,a_{15}$ we define $T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}$. Find the smallest $n$ such that $T(n,a_1,a_2,...,a_{15})$ is divisible by $15$ for any choice of $a_1,a_2,...,a_{15}$.

1994 Austrian-Polish Competition, 7

Tags: number theory , divide , divisible

Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.

1980 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , divide , divisible

Prove that for every nonnegative integer $ k$ there is a product $$(k + 1)(k + 2)...(k + 1980)$$ divisible by $ 1980^{197}$.

1908 Eotvos Mathematical Competition, 1

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

2016 Ecuador NMO (OMEC), 6

Tags: number theory , divisible

A positive integer $n$ is "[i]olympic[/i]" if there are $n$ non-negative integers $x_1, x_2, ..., x_n$ that satisfy that: $\bullet$ There is at least one positive integer $j$: $1 \le j \le n$ such that $x_j \ne 0$. $\bullet$ For any way of choosing $n$ numbers $c_1, c_2, ..., c_n$ from the set $\{-1, 0, 1\}$, where not all $c_i$ are equal to zero, it is true that the sum $c_1x_1 + c_2x_2 +... + c_nx_n$ is not divisible by $n^3$. Find the largest positive "olympic" integer.

2017 Irish Math Olympiad, 1

Tags: coprime , divisible , number theory

Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?

VII Soros Olympiad 2000 - 01, 8.3

Tags: number theory , divisible , algebra

Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.

2001 Kazakhstan National Olympiad, 1

Tags: divide , divisible , number theory

Prove that there are infinitely many natural numbers $ n $ such that $ 2 ^ n + 3 ^ n $ is divisible by $ n $.

2012 Tournament of Towns, 3

Tags: number theory , divide , polynomial , divisible

Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.

2015 Saudi Arabia GMO TST, 4

Tags: number theory , divide , divisible

Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$. Malik Talbi

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 $150$.