Olympiad problems

2023 Chile Junior Math Olympiad, 2

Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.

2002 Estonia National Olympiad, 1

Tags: gcd , lcm , divisible , divide , number theory

The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.

2007 Switzerland - Final Round, 9

Tags: number theory , integer , divide , divisible

Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.

2003 May Olympiad, 3

Tags: number theory , divide , multiple

Find all pairs of positive integers $(a,b)$ such that $8b+1$ is a multiple of $a$ and $8a+1$ is a multiple of $b$.

2016 Dutch Mathematical Olympiad, 3

Tags: greatest common divisor , gcd , number theory , divisor , divide

Find all possible triples $(a, b, c)$ of positive integers with the following properties: • $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$, • $a$ is a divisor of $a + b + c$, • $b$ is a divisor of $a + b + c$, • $c$ is a divisor of $a + b + c$. (Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)

2012 China Northern MO, 3

Tags: number theory , divide

Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that: (1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$ (2) If $m,n \in S$ , then $mn\in S$.

2002 Singapore Team Selection Test, 3

Tags: divide , number theory

For every positive integer $n$, show that there is a positive integer $k$ such that $2k^2 + 2001k + 3 \equiv 0$ (mod $2^n$).

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 $.

1927 Eotvos Mathematical Competition, 1

Tags: number theory , multiple , divide

Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$ Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.

2011 QEDMO 9th, 4

Tags: factorial , number theory , multiple , divide

Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

1925 Eotvos Mathematical Competition, 1

Tags: number theory , divide , divisible

Let $a,b, c,d$ be four integers. Prove that the product of the six differences $$b - a,c - a,d - a,d - c,d - b, c - b$$ is divisible by $12$.

1959 Poland - Second Round, 4

Tags: number theory , perfect cube , 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.

2008 Thailand Mathematical Olympiad, 4

Tags: number theory , binomial , divisible , divide

Let $n$ be a positive integer. Show that $${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.

2017 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , divide

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 $m^{2016}+m^{2015} +\dots+m^2+m+1$.

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 $(abc + 1)^2$.

1989 Tournament Of Towns, (217) 1

Tags: number theory , divide , divisible , digit

Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers. ( M . N . Gusarov, Leningrad)

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 $10^6 n-1$.

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 $2n - 1$.

1977 Dutch Mathematical Olympiad, 3

Tags: number theory , divide , divisible , 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 $7$.

2003 Switzerland Team Selection Test, 9

Tags: number theory , combinatorics , divide

Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$

2017 Switzerland - Final Round, 4

Tags: number theory , divide , divisible

Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold: $$pq | n^p + 2$$ $$n + 2 | n^p + q^p.$$ Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.

VMEO III 2006 Shortlist, N4

Tags: divisible , digit , number theory , divide

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 $n$.

2000 Tournament Of Towns, 5

Tags: consecutive , number theory , divide , divisible

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)

1997 Singapore MO Open, 2

Tags: binomial , divide , divisible , number theory

Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.

2006 QEDMO 2nd, 5

Tags: number theory , divide

For any natural number $m$, we denote by $\phi (m)$ the number of integers $k$ relatively prime to $m$ and satisfying $1 \le k \le m$. Determine all positive integers $n$ such that for every integer $k > n^2$, we have $n | \phi (nk + 1)$. (Daniel Harrer)