Olympiad problems

2013 Poland - Second Round, 1

Let $b$, $c$ be integers and $f(x) = x^2 + bx + c$ be a trinomial. Prove, that if for integers $k_1$, $k_2$ and $k_3$ values of $f(k_1)$, $f(k_2)$ and $f(k_3)$ are divisible by integer $n \neq 0$, then product $(k_1 - k_2)(k_2 - k_3)(k_3 - k_1)$ is divisible by $n$ too.

1975 IMO Shortlist, 6

Tags: modular arithmetic , Divisibility , Digits , decimal representation , IMO , IMO 1975 , digit sum

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

2021 Azerbaijan IMO 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]

2019 Polish MO Finals, 2

Tags: number theory , prime numbers , Divisibility

Let $p$ a prime number and $r$ an integer such that $p|r^7-1$. Prove that if there exist integers $a, b$ such that $p|r+1-a^2$ and $p|r^2+1-b^2$, then there exist an integer $c$ such that $p|r^3+1-c^2$.

2017 Iran MO (3rd round), 1

Tags: number theory , Divisibility , prime numbers , prime , Iran , IranMO

Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

2015 Cono Sur Olympiad, 1

Tags: Divisibility , number theory

Show that, for any integer $n$, the number $n^3 - 9n + 27$ is not divisible by $81$.

2004 IMO Shortlist, 7

Tags: analytic geometry , Triangle , coordinate geometry , Divisibility , IMO Shortlist

Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

2017 Thailand TSTST, 2

Tags: number theory , Divisibility

$\text{(i)}$ Does there exist a positive integer $m > 2016^{2016}$ such that $\frac{2016^m-m^{2016}}{m+2016}$ is a positive integer? $\text{(ii)}$ Does there exist a positive integer $m > 2017^{2017}$ such that $\frac{2017^m-m^{2017}}{m+2017}$ is a positive integer? [i](Serbia MO 2016 P1)[/i]

1999 Mongolian Mathematical Olympiad, Problem 1

Tags: number theory , Divisibility

Prove that for any positive integer $k$ there exist infinitely many positive integers $m$ such that $3^k\mid m^3+10$.

1968 IMO Shortlist, 21

Tags: number theory , Divisibility , IMO Shortlist

Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that \[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]

1994 IMO Shortlist, 6

Tags: modular arithmetic , number theory , Integer sequence , recurrence relation , Divisibility , IMO Shortlist , Kazakhstan NMO 2022 p6

Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?

2006 France Team Selection Test, 3

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]

2014 Irish Math Olympiad, 2

Tags: number theory , number theory unsolved , Divisibility , induction

Prove that for $N>1$ that $(N^{2})^{2014} - (N^{11})^{106}$ is divisible by $N^6 + N^3 +1$ Is this just a proof by induction or is there a more elegant method? I don't think calculating $N = 2$ was expected.

2018 Israel Olympic Revenge, 1

Tags: prime numbers , Divisibility , number theory , Divisors

Let $n$ be a positive integer. Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$

2000 IMO Shortlist, 4

Tags: algebra , polynomial , number theory , Divisibility , IMO Shortlist , Zsigmondy

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

2017 Azerbaijan BMO TST, 2

Tags: Divisibility , RMN

Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.

2007 IMO Shortlist, 6

Tags: quadratics , number theory , IMO , IMO 2007 , Vieta Jumping , Divisibility

Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 \minus{} 1)^2$ has a positive divisor of the form $ 8kn \minus{} 1$ if and only if $ k$ is even. [url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.[/url] [i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]

2022 Germany Team Selection Test, 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 $$n=\frac{ab+3b+8}{a^2+b+3}.$$

1998 IMO Shortlist, 1

Tags: number theory , Divisibility , polynomial , IMO , IMO 1998 , david monk , Hi

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$.

1984 IMO Shortlist, 16

Tags: number theory , equation , Divisibility , power of 2 , IMO , IMO 1984 , Hi

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

1990 IMO Longlists, 65

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.

2000 Saint Petersburg Mathematical Olympiad, 10.7

Tags: number theory , prime numbers , Divisibility

We'll call a positive integer "almost prime", if it is not divisible by any prime from the interval $[3,19]$. We'll call a number "very non-prime", if it has at least 2 primes from interval $[3,19]$ dividing it. What is the greatest amount of almost prime numbers can be selected, such that the sum of any two of them is a very non-prime number? [I]Proposed by S. Berlov, S. Ivanov[/i]

1962 IMO, 1

Tags: number theory , decimal representation , Divisibility , IMO , IMO 1962

Find the smallest natural number $n$ which has the following properties: a) Its decimal representation has a 6 as the last digit. b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

1988 IMO Shortlist, 7

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

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.$

2005 India IMO Training Camp, 2

Tags: function , number theory , algebra , Divisibility , functional equation , IMO Shortlist

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]