Olympiad problems

2019 IFYM, Sozopol, 8

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

2000 Tournament Of Towns, 4

Tags: cube , combinatorics , Divisibility

Can one place positive integers at all vertices of a cube in such a way that for every pair of numbers connected by an edge, one will be divisible by the other , and there are no other pairs of numbers with this property? (A Shapovalov)

1960 IMO, 1

Tags: algebra , number theory , Divisibility , decimal representation , IMO , IMO 1960 , IMO Shortlist

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

2018 Dutch BxMO TST, 3

Tags: number theory , prime numbers , Divisibility

Let $p$ be a prime number. Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.

2016 India IMO Training Camp, 2

Tags: IMO Shortlist , number theory , Divisibility , Hi

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

2021 Science ON all problems, 1

Tags: algebra , number theory , Divisibility

Find all sequences of positive integers $(a_n)_{n\ge 1}$ which satisfy $$a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1)$$ for all $n\in \mathbb{Z}_{\ge 1}$. [i](Bogdan Blaga)[/i]

1966 IMO Longlists, 42

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

Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by $n.$ [b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$

2019 China Team Selection Test, 4

Tags: number theory , Divisibility , China TST , Lte

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?

2009 Brazil Team Selection Test, 3

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]

2018 India National Olympiad, 4

Tags: algebra , Polynomials , Divisibility , Roots of a polynomial

Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.

2015 IFYM, Sozopol, 5

Tags: number theory , prime divisors , prime numbers , Divisibility

Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?

1990 IMO Shortlist, 20

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.

2007 Gheorghe Vranceanu, 1

Tags: induction , algebra , number theory , Divisibility

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of integers defined recursively as $ x_{n+2}=5x_{n+1}-x_n. $ Prove that $ \left( x_n\right)_{n\ge 1} $ has a subsequence whose terms are multiples of $ 22 $ if $ \left( x_n\right)_{n\ge 1} $ has a term that is multiple of $ 22. $

1969 IMO Longlists, 28

Tags: polynomial , number theory , Sequence , Divisibility , recurrence relation , IMO Shortlist , IMO Longlist

$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$

2021 Cyprus JBMO TST, 1

Tags: Divisibility , number theory

Find all positive integers $n$, such that the number \[ \frac{n^{2021}+101}{n^2+n+1}\] is an integer.

2016 IMO Shortlist, N6

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]

2008 Federal Competition For Advanced Students, P1, 1

Tags: number theory , binomial coefficients , Divisibility

What is the remainder of the number $1 \binom{2008}{0 }+2\binom{2008}{1}+ ...+2009\binom{2008}{2008}$ when divided by $2008$?

2021 Science ON all problems, 2

Tags: number theory , Divisibility , Order

Find all pairs $(p,q)$ of prime numbers such that $$p^q-4~|~q^p-1.$$ [i](Vlad Robu)[/i]

2014 JBMO TST - Macedonia, 3

Tags: number theory , Digits , Divisibility

Find all positive integers $n$ which are divisible by 11 and satisfy the following condition: all the numbers which are generated by an arbitrary rearrangement of the digits of $n$, are also divisible by 11.

2015 IMO Shortlist, N2

Tags: factorial , number theory , IMO Shortlist , Divisibility

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

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]