Olympiad problems

2019 China Team Selection Test, 4

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?

2005 China Team Selection Test, 1

Tags: number theory , prime number , divisibility , power of 2 , prime divisor

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.

1969 IMO Shortlist, 25

Tags: number theory , divisibility , frobenius , additive number theory

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

2015 Lusophon Mathematical Olympiad, 6

Tags: number theory , divisibility , sequence

Let $(a_n)$ be defined by: $$ a_1 = 2, \qquad a_{n+1} = a_n^3 - a_n + 1 $$ Consider positive integers $n,p$, where $p$ is an odd prime. Prove that if $p | a_n$, then $p > n$.

2021 Iran RMM TST, 1

Tags: divisibility , integer polynomial , number theory

Let $P(x)=x^{2016}+2x^{2015}+...+2017,Q(x)=1399x^{1398}+...+2x+1$. Prove that there are strictly increasing sequances $a_i,b_i, i=1,...$ of positive integers such that $gcd(a_i,a_{i+1})=1$ for each $i$. Moreover, for each even $i$, $P(b_i) \nmid a_i, Q(b_i) | a_i$ and for each odd $i$, $P(b_i)|a_i,Q(b_i) \nmid a_i$ Proposed by [i]Shayan Talaei[/i]

2014 Contests, 3

Tags: quadratic residue , divisibility , hensel , surjective

Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$. [i]Warut Suksompong, Thailand[/i]

2022 China Team Selection Test, 6

Tags: number theory , divisibility , divisor

Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that \[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]

2015 Balkan MO Shortlist, N4

Tags: number theory , relatively prime , divisibility

Find all pairs of positive integers $(x,y)$ with the following property: If $a,b$ are relative prime and positive divisors of $ x^3 + y^3$, then $a+b - 1$ is divisor of $x^3+y^3$. (Cyprus)

2024 Kyiv City MO Round 2, Problem 2

Tags: number theory , divisibility

Mykhailo wants to arrange all positive integers from $1$ to $2024$ in a circle so that each number is used exactly once and for any three consecutive numbers $a, b, c$ the number $a + c$ is divisible by $b + 1$. Can he do it? [i]Proposed by Fedir Yudin[/i]

2017 Bundeswettbewerb Mathematik, 1

Tags: blackboard , divisible , divisibility , combinatorics , number theory

The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$, then Amelie wins. Else Boris wins. Who can force a victory?

2015 India Regional MathematicaI Olympiad, 3

Tags: polynomial , integer polynomial , divisibility , algebra

Let $P(x)$ be a polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$. [hide]One additional condition must be given that $P$ is non-constant, which even though is understood.[/hide]

1983 IMO Longlists, 9

Tags: algebra , sequence , divisibility , combinatorics , partial order

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

the 9th XMO, 3

Tags: number theory , divisibility

A sequence $\{a_n\} $ satisfies $a_1$ is a positive integer and $a_{n+1}$ is the largest odd integer that divides $2^n-1+a_n$ for all $n\geqslant 1$. Given a positive integer $r$ which is greater than $1$. Is it possible that there exists infinitely many pairs of ordered positive integers $(m,n)$ for which $m>n$ and $a_m = ra_n$? In other words, if you successfully find [b]an[/b] $a_1$ that yields infinitely many pairs of $(m,n)$ which work fine, you win and the answer is YES. Otherwise you have to proof NO for every possible $a_1$. @below, XMO stands for Xueersi Mathematical Olympiad, where Xueersi (学而思) is a famous tutoring camp in China.

2014 India Regional Mathematical Olympiad, 6

Tags: combinatorics , number theory , divisibility

In the adjacent figure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3$?

1990 IMO Longlists, 79

Tags: modular arithmetic , number theory , divisibility , orders and lte

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

2005 IMO Shortlist, 6

Tags: modular arithmetic , number theory , divisibility , chinese remainder theorem

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

1979 IMO, 1

Tags: number theory , relatively prime , series summation , divisibility , prime

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

2010 Brazil Team Selection Test, 2

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

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]

2016 Peru IMO TST, 16

Tags: polynomial , divisibility , divisible , algebra , positive integer

Find all pairs $ (m, n)$ of positive integers that have the following property: For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.

2008 India Regional Mathematical Olympiad, 5

Tags: number theory , divisibility , positive integer

Let $N$ be a ten digit positive integer divisible by $7$. Suppose the first and the last digit of $N$ are interchanged and the resulting number (not necessarily ten digit) is also divisible by $7$ then we say that $N$ is a good integer. How many ten digit good integers are there?

2010 Postal Coaching, 1

Tags: combinatorics , permutation , divisibility

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]

2025 Bulgarian Winter Tournament, 12.3

Tags: function , sequence , divisibility , polynomial , number theory , large number

Determine all functions $f: \mathbb{Z}_{\geq 2025} \to \mathbb{Z}_{>0}$ such that $mn+1$ divides $f(m)f(n) + 1$ for any integers $m,n \geq 2025$ and there exists a polynomial $P$ with integer coefficients, such that $f(n) \leq P(n)$ for all $n\geq 2025$.

2009 IMO Shortlist, 4

Tags: modular arithmetic , number theory , sequence , divisibility

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

2017 Ukraine Team Selection Test, 2

Tags: number theory , function , divisibility

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]

1988 IMO Shortlist, 7

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

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$