Olympiad problems

2019 IFYM, Sozopol, 8

Find all polynomials $f\in Z[X],$ such that for each odd prime $p$ $$f(p)|(p-3)!+\frac{p+1}{2}.$$

2016 Kyiv Mathematical Festival, P5

Tags: combinatorics , digit , divisibility

On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?

2021 South Africa National Olympiad, 1

Tags: number theory , divisibility

Find the smallest and largest integers with decimal representation of the form $ababa$ ($a \neq 0$) that are divisible by $11$.

2015 Czech and Slovak Olympiad III A, 6

Tags: divisibility , number theory , combinatorics , pigeonhole principle

Integer $n>2$ is given. Find the biggest integer $d$, for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$.

2023 Ukraine National Mathematical Olympiad, 9.2

Tags: number theory , divisibility

Positive integers $a_1, a_2, \ldots, a_{101}$ are such that $a_i+1$ is divisible by $a_{i+1}$ for all $1 \le i \le 101$, where $a_{102} = a_1$. What is the largest possible value of $\max(a_1, a_2, \ldots, a_{101})$? [i]Proposed by Oleksiy Masalitin[/i]

2010 Germany Team Selection Test, 1

Tags: function , algebra , modular arithmetic , divisibility , number theory

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

2023 Ukraine National Mathematical Olympiad, 11.1

Tags: number theory , divisibility

Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$? [i]Proposed by Oleksiy Masalitin[/i]

2009 Germany Team Selection Test, 2

Tags: number theory , modular arithmetic , sequence , divisibility

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2012 Belarus Team Selection Test, 1

Tags: algorithm , number theory , prime number , divisibility , number of divisors

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

2002 USAMO, 5

Tags: algorithm , divisibility

Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).

2009 Brazil Team Selection Test, 3

Tags: number theory , modular arithmetic , sequence , divisibility

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

1994 IMO Shortlist, 7

Tags: number theory , decimal representation , divisibility

A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.

2000 IMO Shortlist, 4

Tags: algebra , polynomial , number theory , divisibility

Find all triplets of positive integers $ (a,m,n)$ such that $ a^m \plus{} 1 \mid (a \plus{} 1)^n$.

1999 IMO Shortlist, 6

Tags: algebra , modular arithmetic , arithmetic sequence , divisibility , sum of digits

Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

2020 Indonesia MO, 3

Tags: algebra , functional equation , divisibility

The wording is just ever so slightly different, however the problem is identical. Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.

2023 Indonesia TST, 1

Tags: divisibility , number theory , prime

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

1993 IMO Shortlist, 3

Tags: number theory , representation , divisibility

Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

1992 IMO Longlists, 30

Tags: number theory , divisibility

Let $P_n = (19 + 92)(19^2 +92^2) \cdots(19^n +92^n)$ for each positive integer $n$. Determine, with proof, the least positive integer $m$, if it exists, for which $P_m$ is divisible by $33^{33}.$

1990 IMO Shortlist, 7

Tags: algebra , polynomial , arithmetic sequence , number theory , functional equation , divisibility

Let $ f(0) \equal{} f(1) \equal{} 0$ and \[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\] Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$

2022 Spain Mathematical Olympiad, 6

Tags: number theory , divisibility

Find all triples $(x,y,z)$ of positive integers, with $z>1$, satisfying simultaneously that \[x\text{ divides }y+1,\quad y\text{ divides }z-1,\quad z\text{ divides }x^2+1.\]