Olympiad problems

2022 Bulgaria JBMO TST, 3

The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$,

2000 Belarus Team Selection Test, 8.2

Tags: modular arithmetic , number theory , Integer sequence , Divisibility , Sequence , IMO Shortlist , Hi

Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

2024 Germany Team Selection Test, 2

Tags: number theory , number theory proposed , Digits , product of digits , Divisibility

Show that there exists a real constant $C>1$ with the following property: For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.) [i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]

2011 Peru IMO TST, 3

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

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

2008 Germany Team Selection Test, 3

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

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

2011 Brazil Team Selection Test, 4

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

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

2019 Poland - Second Round, 4

Tags: number theory , positive integers , Divisibility

Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

2012 APMO, 3

Tags: modular arithmetic , number theory , prime , Divisibility

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

2015 Harvard-MIT Mathematics Tournament, 9

Tags: Divisibility , algebra , number theory

Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.

2022 European Mathematical Cup, 2

Tags: number theory , Divisors , greatest common divisor , Divisibility

We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$. a) Are there infinitely many lovely numbers? b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?

2022 Taiwan TST Round 1, 2

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

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2022 Thailand TSTST, 2

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

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2020 China Northern MO, BP3

Tags: number theory , Divisibility , Existence

Are there infinitely many positive integers $n$ such that $19|1+2^n+3^n+4^n$? Justify your claim.

2022 Switzerland Team Selection Test, 1

Tags: number theory , Divisibility , number bases , Switzerland TST

Let $n$ be a positive integer. Prove that there exists a finite sequence $S$ consisting of only zeros and ones, satisfying the following property: for any positive integer $d \geq 2$, when $S$ is interpreted in base $d$, the resulting number is non-zero and divisible by $n$. [i]Remark: The sequence $S=s_ks_{k-1} \cdots s_1s_0$ interpreted in base $d$ is the number $\sum_{i=0}^{k}s_id^i$[/i]

2005 IMO Shortlist, 6

Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist , Chinese Remainder Theorem , Hi

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]

2022 Greece Team Selection Test, 1

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

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2003 Federal Math Competition of S&M, Problem 1

Tags: number theory , floor function , Divisibility , recursion

Prove that the number $\left\lfloor\left(5+\sqrt{35}\right)^{2n-1}\right\rfloor$ is divisible by $10^n$ for each $n\in\mathbb N$.

2010 Brazil Team Selection Test, 2

Tags: function , algebra , modular arithmetic , number theory , Divisibility , IMO Shortlist

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]

1977 IMO Longlists, 10

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 $q^2 + r = 1977.$

1962 All-Soviet Union Olympiad, 12

Tags: number theory , Russia , algebra , Divisibility

Given unequal integers $x, y, z$ prove that $(x-y)^5 + (y-z)^5 + (z-x)^5$ is divisible by $5(x-y)(y- z)(z-x)$.

2022 Kazakhstan National Olympiad, 6

Tags: Sequence , Divisibility , number theory

Given an infinite positive integer sequence $\{x_i\}$ such that $$x_{n+2}=x_nx_{n+1}+1$$ Prove that for any positive integer $i$ there exists a positive integer $j$ such that $x_j^j$ is divisible by $x_i^i$. [i]Remark: Unfortunately, there was a mistake in the problem statement during the contest itself. In the last sentence, it should say "for any positive integer $i>1$ ..."[/i]

2022 Bulgaria JBMO TST, 3

2000 Belarus Team Selection Test, 8.2

2024 Germany Team Selection Test, 2

2011 Peru IMO TST, 3

2008 Germany Team Selection Test, 3

2011 Brazil Team Selection Test, 4

2019 Poland - Second Round, 4

2012 APMO, 3

2015 Harvard-MIT Mathematics Tournament, 9

2022 European Mathematical Cup, 2

2022 Taiwan TST Round 1, 2

2022 Thailand TSTST, 2

2020 China Northern MO, BP3

2022 Switzerland Team Selection Test, 1

2005 IMO Shortlist, 6

2022 Greece Team Selection Test, 1

2003 Federal Math Competition of S&M, Problem 1

2010 Brazil Team Selection Test, 2

1977 IMO Longlists, 10

1962 All-Soviet Union Olympiad, 12

2022 Kazakhstan National Olympiad, 6

1977 IMO, 2

2007 Nicolae Coculescu, 3

1982 IMO Shortlist, 16

2015 Danube Mathematical Competition, 3