Olympiad problems

1983 IMO Longlists, 26

Let $a, b, c$ be positive integers satisfying $\gcd (a, b) = \gcd (b, c) = \gcd (c, a) = 1$. Show that $2abc-ab-bc-ca$ cannot be represented as $bcx+cay +abz$ with nonnegative integers $x, y, z.$

2010 Germany Team Selection Test, 3

Tags: induction , Diophantine equation , number theory unsolved , number theory

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

2001 Austrian-Polish Competition, 7

Tags: number theory unsolved , number theory

Consider the set $A$ containing all positive integers whose decimal expansion contains no $0$, and whose sum $S(N)$ of the digits divides $N$. (a) Prove that there exist infinitely many elements in $A$ whose decimal expansion contains each digit the same number of times as each other digit. (b) Explain that for each positive integer $k$ there exist an element in $A$ having exactly $k$ digits.

2010 India IMO Training Camp, 8

Tags: number theory , prime numbers , number theory unsolved

Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers. (a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.

2014 Romania Team Selection Test, 4

Tags: function , induction , number theory , relatively prime , prime numbers , number theory unsolved

Let $f$ be the function of the set of positive integers into itself, defined by $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(n) + f(n + 1)$. Show that, for any positive integer $n$, the number of positive odd integers m such that $f(m) = n$ is equal to the number of positive integers[color=#0000FF][b] less or equal to [/b][/color]$n$ and coprime to $n$. [color=#FF0000][mod: the initial statement said less than $n$, which is wrong.][/color]

2012 Kyrgyzstan National Olympiad, 5

Tags: modular arithmetic , number theory unsolved , number theory

The sequence of natural numbers is defined as follows: for any $ k\geq 1 $,$ a_{k+2}= a_{k+1}\cdot a_k+1 $. Prove that for $ k\geq 9 $ the number $ a_k-22 $ is composite.

2002 Germany Team Selection Test, 1

Tags: quadratics , number theory unsolved , number theory

Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.

2009 Argentina Iberoamerican TST, 1

Tags: number theory unsolved , number theory

Find all positive integers $ (x,y)$ such that $ \frac{y^2x}{x\plus{}y}$ is a prime number

2008 Baltic Way, 9

Tags: LaTeX , Euler , number theory unsolved , number theory

Suppose that the positive integers $ a$ and $ b$ satisfy the equation $ a^b\minus{}b^a\equal{}1008$ Prove that $ a$ and $ b$ are congruent modulo 1008.

2000 Brazil Team Selection Test, Problem 4

Tags: inequalities , number theory , relatively prime , number theory unsolved

[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)

2005 Germany Team Selection Test, 1

Tags: Putnam , inequalities , number theory unsolved , number theory

Find the smallest positive integer $n$ with the following property: For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that \[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]

2002 Irish Math Olympiad, 3

Tags: number theory unsolved , number theory

Find all triples of positive integers $ (p,q,n)$, with $ p$ and $ q$ primes, satisfying: $ p(p\plus{}3)\plus{}q(q\plus{}3)\equal{}n(n\plus{}3)$.

2012 South East Mathematical Olympiad, 3

Tags: number theory unsolved , number theory

For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).

2005 Irish Math Olympiad, 4

Tags: modular arithmetic , number theory unsolved , number theory

Find the first digit to the left and the first digit to the right of the decimal point in the expansion of $ (\sqrt{2}\plus{}\sqrt{5})^{2000}.$

1993 China Team Selection Test, 1

Tags: modular arithmetic , number theory unsolved , number theory

For all primes $p \geq 3,$ define $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$ and $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$, where $\{x\} = x - [x],$ find the value of $f(p).$

2014 Saudi Arabia IMO TST, 1

Tags: number theory unsolved , number theory

A [i]perfect number[/i] is an integer that equals half the sum of its positive divisors. For example, because $2 \cdot 28 = 1 + 2 + 4 + 7 + 14 + 28$, $28$ is a perfect number. [list] [*] [b](a)[/b] A [i]square-free[/i] integer is an integer not divisible by a square of any prime number. Find all square-free integers that are perfect numbers. [*] [b](b)[/b] Prove that no perfect square is a perfect number.[/list]

2012 Baltic Way, 7

Tags: modular arithmetic , analytic geometry , number theory unsolved , number theory

On a $2012 \times 2012$ board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the marked cells are 0's. Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour. Prove that the number in the bottom right corner is not divisible by 2011.

2012 CentroAmerican, 2

Tags: induction , number theory unsolved , number theory

Alexander and Louise are a pair of burglars. Every morning, Louise steals one third of Alexander's money, but feels remorse later in the afternoon and gives him half of all the money she has. If Louise has no money at the beginning and starts stealing on the first day, what is the least positive integer amount of money Alexander must have so that at the end of the 2012th day they both have an integer amount of money?

2000 Macedonia National Olympiad, 4

Tags: modular arithmetic , number theory , number theory unsolved

Let $a,b$ be coprime positive integers. Show that the number of positive integers $n$ for which the equation $ax+by=n$ has no positive integer solutions is equal to $\frac{(a-1)(b-1)}{2}-1$.

2004 APMO, 1

Tags: induction , greatest common divisor , number theory unsolved , number theory

Determine all finite nonempty sets $S$ of positive integers satisfying \[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \] where $(i,j)$ is the greatest common divisor of $i$ and $j$.

2011 Korea National Olympiad, 2

Tags: modular arithmetic , number theory , greatest common divisor , number theory unsolved

Let $x, y$ be positive integers such that $\gcd(x,y)=1$ and $x+3y^2$ is a perfect square. Prove that $x^2+9y^4$ can't be a perfect square.

2009 Princeton University Math Competition, 2

Tags: number theory unsolved , number theory

Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$.

1983 IMO Longlists, 55

Tags: number theory unsolved , number theory

For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set \[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\] Find $\max_{a\leq 1983} M(a).$

2002 Romania Team Selection Test, 1

Tags: induction , number theory unsolved , number theory

Find all sets $A$ and $B$ that satisfy the following conditions: a) $A \cup B= \mathbb{Z}$; b) if $x \in A$ then $x-1 \in B$; c) if $x,y \in B$ then $x+y \in A$. [i]Laurentiu Panaitopol[/i]

2010 Indonesia TST, 3

Tags: number theory , least common multiple , function , relatively prime , number theory unsolved

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $