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 $10 | a^3b - ab^3$.

2020 Regional Olympiad of Mexico Northeast, 4

Tags: number theory , divide , prime , perfect square

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.

2014 Saudi Arabia GMO TST, 2

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

Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$

1939 Eotvos Mathematical Competition, 2

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

Determine the highest power of $2$ that divides $2^n!$.

2002 Kazakhstan National Olympiad, 7

Tags: prime divisor , divisor , divide , number theory

Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

2017 Puerto Rico Team Selection Test, 3

Tags: multiple , divide , number theory

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$.

2022 New Zealand MO, 5

Tags: number theory , multiple , divide , recurrence relation , divisible

The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.

2014 Regional Olympiad of Mexico Center Zone, 1

Tags: number theory , divisible , divide , product

Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.

2017 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: divide , number theory

Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that $m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.

2013 Saudi Arabia Pre-TST, 2.1

Tags: number theory , remainder , divide , divisible

Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.

2012 QEDMO 11th, 12

Tags: number theory , divisor , divide , divisible

Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.

2001 Kazakhstan National Olympiad, 1

Tags: divisible , divide , number theory

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

2015 Saudi Arabia Pre-TST, 3.3

Tags: number theory , divide

Let $(a_n)_{n\ge0}$ be a sequence of positive integers such that $a^2_n$ divides $a_{n-1}a_{n+1}$, for all $n \ge 1$. Prove that if there exists an integer $k \ge 2$ such that $a_k$ and $a_1$ are relatively prime, then $a_1$ divides $a_0$. (Malik Talbi)

2018 Grand Duchy of Lithuania, 4

Tags: divide , divisor , number theory

Find all positive integers $n$ for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

1977 Dutch Mathematical Olympiad, 3

Tags: divisible , number theory , divide , multiple

From each set $ \{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.

VMEO III 2006 Shortlist, N4

Tags: divisible , digit , divide , number theory

Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.

2008 Dutch Mathematical Olympiad, 3

Tags: number theory , divisible , divide , combinatorics

Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included). Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.

2018 Puerto Rico Team Selection Test, 3

Tags: number theory , divide , divisible

Let $A$ be a set of $m$ positive integers where $m\ge 1$. Show that there exists a nonempty subset $B$ of $A$ such that the sum of all the elements of $B$ is divisible by $m$.

2009 China Northern MO, 8

Tags: number theory , sum of digits , divide

Find the smallest positive integer $N$ satisfies : 1 . $209$│$N$ 2 . $ S (N) = 209 $ ( # Here $S(m)$ means the sum of digits of number $m$ )

2012 NZMOC Camp Selection Problems, 2

Tags: number theory , divisible , consecutive , divide , sum

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

2005 iTest, 25

Tags: number theory , divide , divisible

Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?

1977 Swedish Mathematical Competition, 1

Tags: divide , divisible , number theory

$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.

2000 Czech And Slovak Olympiad IIIA, 1

Tags: divisible , number theory , even , divide

Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.

2012 India Regional Mathematical Olympiad, 2

Tags: divisibility , divide , divisor , number theory

Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.

2017 Latvia Baltic Way TST, 16

Tags: divisible , combinatorics , divide , number theory

Strings $a_1, a_2, ... , a_{2016}$ and $b_1, b_2, ... , b_{2016}$ each contain all natural numbers from $1$ to $2016$ exactly once each (in other words, they are both permutations of the numbers $1, 2, ..., 2016$). Prove that different indices $i$ and $j$ can be found such that $a_ib_i- a_jb_j$ is divisible by $2017$.