Olympiad problems

2022 Bundeswettbewerb Mathematik, 4

For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that: (a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$. (b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.

1996 Irish Math Olympiad, 1

Tags: number theory proposed , number theory

The Fibonacci sequence is defined by $ F_0\equal{}0, F_1\equal{}1$ and $ F_{n\plus{}2}\equal{}F_n\plus{}F_{n\plus{}1}$ for $ n \ge 0$. Prove that: $ (a)$ The statement $ "F_{n\plus{}k}\minus{}F_n$ is divisible by $ 10$ for all $ n \in \mathbb{N}"$ is true if $ k\equal{}60$ but false for any positive integer $ k<60$. $ (b)$ The statement $ "F_{n\plus{}t}\minus{}F_n$ is divisible by $ 100$ for all $ n \in \mathbb{N}"$ is true if $ t\equal{}300$ but false for any positive integer $ t<300$.

1970 Miklós Schweitzer, 4

Tags: modular arithmetic , quadratics , function , absolute value , number theory proposed , number theory

If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] J. Suranyi

2009 All-Russian Olympiad Regional Round, 10.5

Tags: number theory proposed , number theory

Positive integer $m$ is such that the sum of decimal digits of $2^m$ equals 8. Can the last digit of $2^m$ be equal 6? (Author: V. Senderov)

1998 Hungary-Israel Binational, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all positive integers $ x$ and $ y$ such that $ 5^{x}-3^{y}= 16$.

2024 Czech-Polish-Slovak Junior Match, 5

Tags: number theory , number theory proposed , sum of digits

For a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive integer $n$ for which $4 \cdot S(n)=3 \cdot S(2n)$.

2010 Contests, 3

Tags: number theory proposed , number theory

For $ n\in\mathbb{N}$, determine the number of natural solutions $ (a,b)$ such that \[ (4a\minus{}b)(4b\minus{}a)\equal{}2010^n\] holds.

2008 Grigore Moisil Intercounty, 4

Tags: number theory proposed , number theory

Let $ n$ be a positive integer, and $ k\leq n\minus{}1$, $ k\in \mathbb{N}$. Denote $ a_k\equal{}k!(1\plus{}\frac12\plus{}\frac13\plus{}\cdots\plus{}\frac1k)$. Prove that the number $ k! \cdot\left[\binom{n\minus{}1}{k}\minus{}(\minus{}1)^k\right]\plus{}(\minus{}1)^k\cdot a_k \cdot n$ is divisible by $ n^2$.

2014 Olympic Revenge, 2

Tags: floor function , limit , number theory proposed , number theory

$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$. $b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.

1993 Baltic Way, 4

Tags: number theory proposed , number theory

Determine all integers $n$ for which \[\sqrt{\frac{25}{2}+\sqrt{\frac{625}{4}-n}}+\sqrt{\frac{25}{2}-\sqrt{\frac{625}{4}-n}}\] is an integer.

2002 Romania Team Selection Test, 1

Tags: number theory proposed , number theory

Let $m,n$ be positive integers of distinct parities and such that $m<n<5m$. Show that there exists a partition with two element subsets of the set $\{ 1,2,3,\ldots ,4mn\}$ such that the sum of numbers in each set is a perfect square. [i]Dinu Șerbănescu[/i]

2003 Italy TST, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$

1998 Hong kong National Olympiad, 3

Tags: number theory , greatest common divisor , relatively prime , number theory proposed

Given $s,t$ are non-zero integers, $(x,y) $ is an integer pair , A transformation is to change pair $(x,y)$ into pair $(x+t,y-s)$ . If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" . (1) Is $(s,t)$ a good pair ? (2) Prove :for any $s$ and $t$ , there exists pair $(x,y)$ which is " a good pair".

2009 Hong kong National Olympiad, 4

Tags: modular arithmetic , number theory proposed , number theory

find all pairs of non-negative integer pairs $(m,n)$,satisfies $107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}$

2007 Kyiv Mathematical Festival, 2

Tags: number theory proposed , number theory

Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$

2001 Irish Math Olympiad, 3

Tags: number theory proposed , number theory

Show that if an odd prime number $ p$ can be expressed in the form $ x^5\minus{}y^5$ for some integers $ x,y,$ then: $ \sqrt{\frac{4p\plus{}1}{5}}\equal{}\frac{v^2\plus{}1}{2}$ for some odd integer $ v$.

2006 Taiwan National Olympiad, 1

Tags: number theory proposed , number theory

Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.

2006 Singapore MO Open, 5

Tags: number theory proposed , number theory

Let $a,b,n$ be positive integers. Prove that $n!$ divides \[b^{n-1}a(a+b)(a+2b)...(a+(n-1)b)\]

1979 IMO Longlists, 18

Tags: number theory proposed , number theory

Show that for no integers $a \geq 1, n \geq 1$ is the sum \[1+\frac{1}{1+a}+\frac{1}{1+2a}+\cdots+\frac{1}{1+na}\] an integer.

2013 China Western Mathematical Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Does there exist any integer $a,b,c$ such that $a^2bc+2,ab^2c+2,abc^2+2$ are perfect squares?

2007 Kazakhstan National Olympiad, 3

Tags: number theory , number theory proposed

Solve in prime numbers the equation $p(p+1)+q(q+1)=r(r+1)$.

2019 Turkey Junior National Olympiad, 1

Tags: number theory , number theory proposed , gcd or factors , power of prime

Solve $2a^2+3a-44=3p^n$ in positive integers where $p$ is a prime.

2003 Tuymaada Olympiad, 2

Tags: modular arithmetic , number theory proposed , number theory

Which number is bigger : the number of positive integers not exceeding 1000000 that can be represented by the form $2x^{2}-3y^{2}$ with integral $x$ and $y$ or that of positive integers not exceeding 1000000 that can be represented by the form $10xy-x^{2}-y^{2}$ with integral $x$ and $y?$ [i]Proposed by A. Golovanov[/i]

2007 Polish MO Finals, 4

Tags: number theory proposed , number theory

4. Given is an integer $n\geq 1$. Find out the number of possible values of products $k \cdot m$, where $k,m$ are integers satisfying $n^{2}\leq k \leq m \leq (n+1)^{2}$.

2014 Iran MO (3rd Round), 5

Tags: ratio , algebra , polynomial , number theory proposed , number theory

Can an infinite set of natural numbers be found, such that for all triplets $(a,b,c)$ of it we have $abc + 1 $ perfect square? (20 points )