Olympiad problems

2011 Saudi Arabia BMO TST, 4

Let $(F_n )_{n\ge o}$ be the sequence of Fibonacci numbers: $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ , for every $n \ge 0$. Prove that for any prime $p \ge 3$, $p$ divides $F_{2p} - F_p$ .

2015 Dutch BxMO/EGMO TST, 1

Tags: number theory , divisor , divide

Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

2013 Saudi Arabia GMO TST, 4

Tags: divisible , divide , number theory

Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.

2012 Switzerland - Final Round, 7

Tags: number theory , divide , divisible

Let $n$ and $k$ be natural numbers such that $n = 3k +2$. Show that the sum of all factors of $n$ is divisible by $3$.

2016 Flanders Math Olympiad, 2

Tags: power , divide , divisor , number theory

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

2019 Auckland Mathematical Olympiad, 2

Tags: divisible , divide , number theory

Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.

2023 Chile Junior Math Olympiad, 2

Tags: divide , number theory

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 $n!$.

1998 Singapore Senior Math Olympiad, 1

Tags: integer , divisible , divide , number theory

Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

1978 Chisinau City MO, 159

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

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.

1984 Bundeswettbewerb Mathematik, 1

Tags: divide , divisible , number theory

The natural numbers $n$ and $z$ are relatively prime and greater than $1$. For $k = 0, 1, 2,..., n - 1$ let $s(k) = 1 + z + z^2 + ...+ z^k.$ Prove that: a) At least one of the numbers $s(k)$ is divisible by $n$. b) If $n$ and $z - 1$ are also coprime, then already one of the numbers $s(k)$ with $k = 0,1, 2,..., n- 2$ is divisible by $n$.

2003 Switzerland Team Selection Test, 4

Tags: number theory , divide , max

Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.

1999 Tournament Of Towns, 3

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

Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

2022 Durer Math Competition Finals, 10

Tags: divide , number theory

The pair of positive integers $(a, b)$ is such that a does not divide $b$, $b$ does not divide a, both numbers are at most $100$, and they have the maximal possible number of common divisors. What is the largest possible value of $a \cdot· b$?

1994 Tournament Of Towns, (417) 5

Tags: digit , divide , divisible , number theory

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)

2010 Saudi Arabia BMO TST, 3

Tags: divide , divisible , number theory

How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?

2016 Dutch Mathematical Olympiad, 3

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

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 $y$.)

2005 Chile National Olympiad, 2

Tags: divide , number theory

Let $p$ be a prime number greater than $2$ and let $m, n$ be integers such that: $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}.$$ Prove that $p$ divides $m$.

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]

2022 Saudi Arabia IMO TST, 1

Tags: recurrence relation , divide , number theory with sequences , number theory

Let $(a_n)$ be the integer sequence which is defined by $a_1= 1$ and $$ a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1.$$ Let $S$ be the set of all primes $p$ such that there exists an index $i$ such that $p|a_i$. Prove that the set $S$ is an infinite set and it is not equal to the set of all primes.

2022 Chile Junior Math Olympiad, 4

Tags: number theory , divide

Let $S$ be the sum of all products $ab$ where $a$ and $b$ are distinct elements of the set $\{1,2,...,46\}$. Prove that $47$ divides $S$.

2016 Dutch BxMO TST, 5

Tags: number theory , divide

Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$

2009 Cuba MO, 9

Tags: divide , number theory

Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$

VMEO IV 2015, 10.3

Tags: divide , divisible , number theory

Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

2016 Saudi Arabia GMO TST, 3

Tags: polynomial , divide , divisor , algebra

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 $n \ge 0$

1947 Moscow Mathematical Olympiad, 136

Tags: parallelogram , combinatorics , divide , 13-gon , convex

Prove that no convex $13$-gon can be cut into parallelograms.