Olympiad problems

2002 Bulgaria National Olympiad, 5

Find all pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$, $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms. [i]Proposed by Oleg Mushkarov and Nikolai Nikolov[/i]

2013 Baltic Way, 19

Tags: algebra , polynomial , modular arithmetic , number theory unsolved , number theory

Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\ge 1$. Can $a_0$ be chosen so that $a_{54}$ is a multiple of $2013$?

2012 India Regional Mathematical Olympiad, 2

Tags: number theory unsolved , number theory

Let $a,b,c$ be positive integers such that $a|b^5, b|c^5$ and $c|a^5$. Prove that $abc|(a+b+c)^{31}$.

2010 China Team Selection Test, 3

Tags: number theory unsolved , number theory

For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.

2011 China Team Selection Test, 3

Tags: number theory , IMO Shortlist , number theory unsolved

For any positive integer $d$, prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number.

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.

2005 Taiwan National Olympiad, 2

Tags: modular arithmetic , number theory unsolved , number theory

$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that $\min{\{x,y,z\}} \le ab+bc+ca$.

2010 Postal Coaching, 3

Tags: number theory unsolved , number theory

Prove that a prime $p$ is expressible in the form $x^2+3y^2;x,y\in Z$ if and only if it is expressible in the form $ m^2+mn+n^2;m,n \in Z$.Can $p$ be replaced by a natural number $n$?

2001 Greece National Olympiad, 2

Tags: modular arithmetic , number theory unsolved , number theory

Prove that there are no positive integers $a,b$ such that $(15a +b)(a +15b)$ is a power of $3.$

2018 Romania Team Selection Tests, 4

Tags: number theory , greatest common divisor , bezout s identity , number theory unsolved

Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.

2014 Dutch IMO TST, 2

Tags: number theory unsolved , number theory

The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$.

2011 Croatia Team Selection Test, 4

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

We define the sequence $x_n$ so that \[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\] Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.

2007 Pre-Preparation Course Examination, 15

Tags: number theory unsolved , number theory

Does there exists a subset of positive integers with infinite members such that for every two members $a,b$ of this set \[a^2-ab+b^2|(ab)^2\]

2006 MOP Homework, 5

Tags: number theory unsolved , number theory

Let $a$, $b$, and $c$ be positive integers such that the product $ab$ divides the product $c(c^2-c+1)$ and the sum $a+b$ is divisible the number $c^2+1$. Prove that the sets ${a,b}$ and ${c,c^2-c+1}$ coincide.

2003 Hong kong National Olympiad, 4

Tags: number theory unsolved , number theory

Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.

2008 Baltic Way, 6

Tags: number theory unsolved , number theory , extremal principle

Find all finite sets of positive integers with at least two elements such that for any two numbers $ a$, $ b$ ($ a > b$) belonging to the set, the number $ \frac {b^2}{a \minus{} b}$ belongs to the set, too.

1993 Irish Math Olympiad, 2

Tags: number theory , prime numbers , number theory unsolved

A positive integer $ n$ is called $ good$ if it can be uniquely written simultaneously as $ a_1\plus{}a_2\plus{}...\plus{}a_k$ and as $ a_1 a_2...a_k$, where $ a_i$ are positive integers and $ k \ge 2$. (For example, $ 10$ is good because $ 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1$ is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.

1987 Kurschak Competition, 1

Tags: number theory unsolved , number theory

Find all quadruples of positive integers $(a,b,c,d)$ such that $a+b=cd$ and $c+d=ab$.

2010 Malaysia National Olympiad, 8

Tags: floor function , number theory unsolved , number theory

For any number $x$, let $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. A sequence $a_1,a_2,\cdots$ is given, where \[a_n=\left\lfloor{\sqrt{2n}+\dfrac{1}{2}}\right\rfloor.\] How many values of $k$ are there such that $a_k=2010$?

2007 Bosnia Herzegovina Team Selection Test, 2

Tags: number theory unsolved , number theory

Find all pairs of integers $(x,y)$ such that $x(x+2)=y^2(y^2+1)$

2014 Dutch IMO TST, 2

Tags: number theory unsolved , number theory

The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$.

2002 Iran Team Selection Test, 9

Tags: function , calculus , integration , inequalities , number theory unsolved , number theory

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

2011 Czech and Slovak Olympiad III A, 2

Tags: number theory unsolved , number theory

Find all triples $(p, q, r)$ of prime numbers for which \[(p+1)(q+2)(r+3)=4pqr. \]

2011 Iran Team Selection Test, 4

Tags: number theory , relatively prime , number theory unsolved

Define a finite set $A$ to be 'good' if it satisfies the following conditions: [list][*][b](a)[/b] For every three disjoint element of $A,$ like $a,b,c$ we have $\gcd(a,b,c)=1;$ [*][b](b)[/b] For every two distinct $b,c\in A,$ there exists an $a\in A,$ distinct from $b,c$ such that $bc$ is divisible by $a.$[/list] Find all good sets.

2014 Contests, 1

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

A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.