Olympiad problems

2015 Cono Sur Olympiad, 5

Determine if there exists an infinite sequence of not necessarily distinct positive integers $a_1, a_2, a_3,\ldots$ such that for any positive integers $m$ and $n$ where $1 \leq m < n$, the number $a_{m+1} + a_{m+2} + \ldots + a_{n}$ is not divisible by $a_1 + a_2 + \ldots + a_m$.

2007 IMO Shortlist, 3

Tags: modular arithmetic , number theory , Divisibility , Extremal combinatorics , Additive combinatorics , IMO Shortlist

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

1997 Romania Team Selection Test, 2

Tags: modular arithmetic , number theory solved , number theory , Divisibility , Multiplicative NT , Multiplicative order

Suppose that $A$ be the set of all positive integer that can write in form $a^2+2b^2$ (where $a,b\in\mathbb {Z}$ and $b$ is not equal to $0$). Show that if $p$ be a prime number and $p^2\in A$ then $p\in A$. [i]Marcel Tena[/i]

2024 Bangladesh Mathematical Olympiad, P9

Tags: number theory , factorial , Divisibility , p-adic

Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.

2014 Contests, 3

Tags: number theory , Divisibility

Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 | (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.

2013 North Korea Team Selection Test, 3

Tags: modular arithmetic , quadratics , number theory , North Korea , TST , Divisibility

Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.

2018 Israel National Olympiad, 4

Tags: number theory , Digits , digit sum , Divisibility

The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

2002 Iran MO (2nd round), 1

Tags: combinatorics , permutation , Divisibility , IMO Shortlist

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

2009 Belarus Team Selection Test, 3

Tags: combinatorics , permutation , Divisibility , IMO Shortlist

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

2023 India Regional Mathematical Olympiad, 1

Tags: number theory , RMO 2023 , Squares , Divisibility

Let $\mathbb{N}$ be the set of all positive integers and $S=\left\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right\}$. Find the largest positive integer $m$ such that $m$ divides abcd for all $(a, b, c, d) \in S$.

1997 Moldova Team Selection Test, 3

Tags: pigeonhole principle , number theory , decimal representation , Divisibility , multiple , IMO Shortlist , Digits

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

1990 IMO Shortlist, 7

Tags: algebra , polynomial , arithmetic sequence , number theory , functional equation , Divisibility , IMO Shortlist

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.$

1977 IMO Shortlist, 3

Tags: number theory , Additive Number Theory , remainder , Divisibility , IMO , IMO 1977

Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

2019 Tournament Of Towns, 2

Tags: number theory , Sum of powers , Divisibility

Consider two positive integers $a$ and $b$ such that $a^{n+1} + b^{n+1}$ is divisible by $a^n + b^n$ for infinitely many positive integers $n$. Is it necessarily true that $a = b$? (Boris Frenkin)

1974 IMO Shortlist, 6

Tags: number theory , Summation , binomial coefficients , Divisibility , modular arithmetic , IMO , IMO 1974

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

2019 ELMO Shortlist, N4

Tags: number theory , Divisibility

A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$. [i]Proposed by Carl Schildkraut and Holden Mui[/i]

1992 IMO Longlists, 30

Tags: number theory , Divisibility , IMO Shortlist , IMO Longlist

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 $33^{33}.$

2023 Romania EGMO TST, P2

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2008 IMO Shortlist, 2

Tags: number theory , modular arithmetic , Sequence , Divisibility , IMO Shortlist

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]

2025 Bangladesh Mathematical Olympiad, P8

Tags: number theory , Divisibility , GCD , modular arithmetic

Let $a, b, m, n$ be positive integers such that $gcd(a, b) = 1$ and $a > 1$. Prove that if $$a^m+b^m \mid a^n+b^n$$then $m \mid n$.

2021 Thailand TST, 1

Tags: number theory , Sequence , Divisibility , IMO Shortlist , IMO Shortlist 2020

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

2017 Taiwan TST Round 2, 3

Tags: number theory , IMO Shortlist , function , Divisibility , Hi

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

IMSC 2024, 6

Tags: algebra , polynomial , number theory , number theory proposed , algebra proposed , Lifting the Exponent , Divisibility

Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that $$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$ is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial. [i]Proposed by Vlad Matei, Romania[/i]

2017 Korea National Olympiad, problem 5

Tags: number theory , polynomial , Cubic , Divisibility

Given a prime $p$, show that there exist two integers $a, b$ which satisfies the following. For all integers $m$, $m^3+ 2017am+b$ is not a multiple of $p$.

2016 Belarus Team Selection Test, 3

Tags: factorial , number theory , IMO Shortlist , Divisibility

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.