Olympiad problems

2005 International Zhautykov Olympiad, 2

Let $ m,n$ be integers such that $ 0\le m\le 2n$. Then prove that the number $ 2^{2n \plus{} 2} \plus{} 2^{m \plus{} 2} \plus{} 1$ is perfect square iff $ m \equal{} n$.

2011 Pre - Vietnam Mathematical Olympiad, 1

Tags: abstract algebra , number theory proposed , number theory

Determine all values of $n$ satisfied the following condition: there's exist a cyclic $(a_1,a_2,a_3,...,a_n)$ of $(1,2,3,...,n)$ such that $\left\{ {{a_1},{a_1}{a_2},{a_1}{a_2}{a_3},...,{a_1}{a_2}...{a_n}} \right\}$ is a complete residue systems modulo $n$.

1989 Balkan MO, 1

Tags: search , number theory proposed , number theory

Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$. Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$.

2008 Polish MO Finals, 6

Tags: number theory proposed , number theory

Let $ S$ be a set of all positive integers which can be represented as $ a^2 \plus{} 5b^2$ for some integers $ a,b$ such that $ a\bot b$. Let $ p$ be a prime number such that $ p \equal{} 4n \plus{} 3$ for some integer $ n$. Show that if for some positive integer $ k$ the number $ kp$ is in $ S$, then $ 2p$ is in $ S$ as well. Here, the notation $ a\bot b$ means that the integers $ a$ and $ b$ are coprime.

1998 Brazil National Olympiad, 1

Tags: number theory , relatively prime , number theory proposed

15 positive integers, all less than 1998(and no one equal to 1), are relatively prime (no pair has a common factor > 1). Show that at least one of them must be prime.

2005 Colombia Team Selection Test, 1

Tags: Columbia , geometry , 3D geometry , number theory proposed , number theory

Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!

2012 ELMO Shortlist, 5

Tags: modular arithmetic , number theory proposed , number theory

Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$. [i]Ravi Jagadeesan.[/i]

2006 Czech and Slovak Olympiad III A, 5

Tags: number theory proposed , number theory

Find all triples $(p,q,r)$ of pairwise distinct primes such that \[p\mid q+r, q\mid r+2p, r\mid p+3q.\]

2002 Moldova Team Selection Test, 1

Tags: inequalities , number theory proposed , number theory

Consider the triangular numbers $T_n = \frac{n(n+1)}{2} , n \in \mathbb N$. [list][b](a)[/b] If $a_n$ is the last digit of $T_n$, show that the sequence $(a_n)$ is periodic and find its basic period. [b](b)[/b] If $s_n$ is the sum of the first $n$ terms of the sequence $(T_n)$, prove that for every $n \geq 3$ there is at least one perfect square between $s_{n-1} and $s_n$.[/list]

1998 Turkey MO (2nd round), 1

Tags: modular arithmetic , quadratics , number theory proposed , number theory

Find all positive integers $x$ and $n$ such that ${{x}^{3}}+3367={{2}^{n}}$.

2006 ISI B.Math Entrance Exam, 8

Tags: number theory proposed

Let $S$ be the set of all integers $k$, $1\leq k\leq n$, such that $\gcd(k,n)=1$. What is the arithmetic mean of the integers in $S$?

2008 Kazakhstan National Olympiad, 1

Tags: inequalities , number theory proposed , number theory , Kazakhstan

Find all integer solutions $ (a_1,a_2,\dots,a_{2008})$ of the following equation: $ (2008\minus{}a_1)^2\plus{}(a_1\minus{}a_2)^2\plus{}\dots\plus{}(a_{2007}\minus{}a_{2008})^2\plus{}a_{2008}^2\equal{}2008$

2009 Portugal MO, 1

Tags: modular arithmetic , number theory proposed , number theory

A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided?

2003 Moldova National Olympiad, 10.1

Tags: number theory proposed , number theory

Find all prime numbers $ a,b,c$ that fulfill the equality: $ (a\minus{}2)!\plus{}2b!\equal{}22c\minus{}1$

2007 Hong kong National Olympiad, 4

Tags: floor function , number theory proposed , number theory

find all positive integer pairs $(m,n)$,satisfies: (1)$gcd(m,n)=1$,and $m\le\ 2007$ (2)for any $k=1,2,...2007$,we have $[\frac{nk}{m}]=[\sqrt{2}k]$

2010 ELMO Shortlist, 2

Tags: modular arithmetic , algebra , difference of squares , special factorizations , number theory proposed , number theory

Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

1983 IMO Longlists, 10

Tags: number theory proposed , number theory

Which of the numbers $1, 2, \ldots , 1983$ has the largest number of divisors?

2014 China Northern MO, 3

Tags: Diophantine equation , number theory proposed , number theory

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

1990 Brazil National Olympiad, 2

Tags: number theory proposed , number theory

There exists infinitely many positive integers such that $a^3 + 1990b^3 = c^4$.

2011 Indonesia TST, 4

Tags: modular arithmetic , number theory proposed , number theory

Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.

2012 European Mathematical Cup, 2

Tags: number theory , greatest common divisor , function , prime numbers , number theory proposed

Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements? [i]Proposed by Ognjen Stipetić.[/i]

2008 Tournament Of Towns, 4

Tags: arithmetic sequence , number theory proposed , number theory

Five distinct positive integers form an arithmetic progression. Can their product be equal to $a^{2008}$ for some positive integer $a$ ?

2009 Ukraine National Mathematical Olympiad, 2

Tags: number theory proposed , number theory

Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$

2013 Stars Of Mathematics, 3

Tags: quadratics , number theory proposed , number theory

Consider the sequence $(a^n + 1)_{n\geq 1}$, with $a>1$ a fixed integer. i) Prove there exist infinitely many primes, each dividing some term of the sequence. ii) Prove there exist infinitely many primes, none dividing any term of the sequence. [i](Dan Schwarz)[/i]

2014 China Team Selection Test, 3

Tags: number theory proposed , number theory , Diophantine equation , China TST

Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$