Olympiad problems

2015 Switzerland Team Selection Test, 8

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

2007 Romania Team Selection Test, 3

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

Let $a_{i}$, $i = 1,2, \dots ,n$, $n \geq 3$, be positive integers, having the greatest common divisor 1, such that \[a_{j}\textrm{ divide }\sum_{i = 1}^{n}a_{i}\] for all $j = 1,2, \dots ,n$. Prove that \[\prod_{i = 1}^{n}a_{i}\textrm{ divides }\Big{(}\sum_{i = 1}^{n}a_{i}\Big{)}^{n-2}.\]

2014 Middle European Mathematical Olympiad, 4

Tags: factorial , algorithm , number theory , relatively prime , number theory proposed

For integers $n \ge k \ge 0$ we define the [i]bibinomial coefficient[/i] $\left( \binom{n}{k} \right)$ by \[ \left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .\] Determine all pairs $(n,k)$ of integers with $n \ge k \ge 0$ such that the corresponding bibinomial coefficient is an integer. [i]Remark: The double factorial $n!!$ is defined to be the product of all even positive integers up to $n$ if $n$ is even and the product of all odd positive integers up to $n$ if $n$ is odd. So e.g. $0!! = 1$, $4!! = 2 \cdot 4 = 8$, and $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$.[/i]

2007 Balkan MO Shortlist, N2

Tags: function , number theory proposed , number theory

Prove that there are no distinct positive integers $x$ and $y$ such that $x^{2007} + y! = y^{2007} + x! $

1982 IMO Longlists, 49

Tags: number theory proposed , number theory

Simplify \[\sum_{k=0}^n \frac{(2n)!}{(k!)^2((n-k)!)^2}.\]

2010 IberoAmerican Olympiad For University Students, 3

Tags: number theory proposed , number theory

A student adds up rational fractions incorrectly: \[\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y}\quad (\star) \] Despite that, he sometimes obtains correct results. For a given fraction $\frac{a}{b},a,b\in\mathbb{Z},b>0$, find all fractions $\frac{x}{y},x,y\in\mathbb{Z},y>0$ such that the result obtained by $(\star)$ is correct.

2005 Georgia Team Selection Test, 7

Tags: number theory proposed , number theory

Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.

2013 Brazil National Olympiad, 4

Tags: number theory proposed , number theory

Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$. P.S.: This is basically the same problem as http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=548550.

2006 JBMO ShortLists, 13

Tags: number theory , greatest common divisor , algorithm , modular arithmetic , number theory proposed

Let $ A$ be a subset of the set $ \{1, 2,\ldots,2006\}$, consisting of $ 1004$ elements. Prove that there exist $ 3$ distinct numbers $ a,b,c\in A$ such that $ gcd(a,b)$: a) divides $ c$ b) doesn't divide $ c$

2007 IberoAmerican Olympiad For University Students, 7

Tags: number theory proposed , number theory

The [i]height[/i] of a positive integer is defined as being the fraction $\frac{s(a)}{a}$, where $s(a)$ is the sum of all the positive divisors of $a$. Show that for every pair of positive integers $N,k$ there is a positive integer $b$ such that the [i]height[/i] of each of $b,b+1,\cdots,b+k$ is greater than $N$.

2009 Moldova Team Selection Test, 2

Tags: ratio , number theory proposed , number theory

[color=darkblue]Let $ M$ be a set of aritmetic progressions with integer terms and ratio bigger than $ 1$. [b]a)[/b] Prove that the set of the integers $ \mathbb{Z}$ can be written as union of the finite number of the progessions from $ M$ with different ratios. [b]b)[/b] Prove that the set of the integers $ \mathbb{Z}$ can not be written as union of the finite number of the progessions from $ M$ with ratios integer numbers, any two of them coprime.[/color]

2009 Pan African, 1

Tags: number theory proposed , number theory

Determine whether or not there exist numbers $x_1,x_2,\ldots ,x_{2009}$ from the set $\{-1,1\}$, such that: \[x_1x_2+x_2x_3+x_3x_4+\ldots+x_{2008}x_{2009}+x_{2009}x_1=999\]

2011 China Team Selection Test, 2

Tags: modular arithmetic , number theory proposed , number theory

Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

2006 Croatia Team Selection Test, 4

Tags: inequalities , number theory proposed , number theory

Find all natural solutions of $3^{x}= 2^{x}y+1.$

1988 Balkan MO, 4

Tags: number theory , prime numbers , number theory proposed

Let $(a_{n})_{n\geq 1}$ be a sequence defined by $a_{n}=2^{n}+49$. Find all values of $n$ such that $a_{n}=pg, a_{n+1}=rs$, where $p,q,r,s$ are prime numbers with $p<q, r<s$ and $q-p=s-r$.

1997 Federal Competition For Advanced Students, Part 2, 1

Tags: number theory proposed , number theory

Let $a$ be a fixed integer. Find all integer solutions $x, y, z$ of the system \[5x + (a + 2)y + (a + 2)z = a,\]\[(2a + 4)x + (a^2 + 3)y + (2a + 2)z = 3a - 1,\]\[(2a + 4)x + (2a + 2)y + (a^2 + 3)z = a + 1.\]

2009 Macedonia National Olympiad, 5

Tags: number theory proposed , number theory

Solve the following equation in the set of integer numbers: \[ x^{2010}-2006=4y^{2009}+4y^{2008}+2007y. \]

1992 Baltic Way, 6

Tags: number theory proposed , number theory

Prove that the product of the 99 numbers $ \frac{k^3\minus{}1}{k^3\plus{}1},k\equal{}2,3,\ldots,100$ is greater than $ 2/3$.

2006 Polish MO Finals, 2

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

Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$.

1995 Polish MO Finals, 3

Tags: induction , modular arithmetic , algebra , polynomial , number theory proposed , number theory

Let $p$ be a prime number, and define a sequence by: $x_i=i$ for $i=,0,1,2...,p-1$ and $x_n=x_{n-1}+x_{n-p}$ for $n \geq p$ Find the remainder when $x_{p^3}$ is divided by $p$.

2002 ITAMO, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all $3$-digit positive integers that are $34$ times the sum of their digits.

2012 India IMO Training Camp, 2

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

Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$

1997 Federal Competition For Advanced Students, Part 2, 2

Tags: modular arithmetic , number theory proposed , number theory

A positive integer $K$ is given. Define the sequence $(a_n)$ by $a_1 = 1$ and $a_n$ is the $n$-th positive integer greater than $a_{n-1}$ which is congruent to $n$ modulo $K$. [b](a)[/b] Find an explicit formula for $a_n$. [b](b)[/b] What is the result if $K = 2$?

2014 Contests, 1

Tags: number theory , prime numbers , number theory proposed

A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

2010 Lithuania National Olympiad, 4

Tags: modular arithmetic , number theory proposed , number theory

Decimal digits $a,b,c$ satisfy \[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \] where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.