Olympiad problems

2011 Postal Coaching, 4

Let $n > 1$ be a positive integer. Find all $n$-tuples $(a_1 , a_2 ,\ldots, a_n )$ of positive integers which are pairwise distinct, pairwise coprime, and such that for each $i$ in the range $1 \le i \le n$, \[(a_1 + a_2 + \ldots + a_n )|(a_1^i + a_2^i + \ldots + a_n^i )\].

1991 China National Olympiad, 4

Tags: number theory unsolved , number theory

Find all positive integer solutions $(x,y,z,n)$ of equation $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$, where $n\ge 2$ and $z \le 5\times 2^{2n}$.

1994 Korea National Olympiad, Problem 1

Tags: number theory unsolved , number theory

Consider the equation $ y^2\minus{}k\equal{}x^3$, where $ k$ is an integer. Prove that the equation cannot have five integer solutions of the form $ (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4)$. Also show that if it has the first four of these pairs as solutions, then $ 63|k\minus{}17$.

2014 Contests, 2

Tags: floor function , number theory unsolved , number theory

Let $n$ be a natural number. Prove that, \[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \] is even.

2008 Ukraine Team Selection Test, 6

Tags: modular arithmetic , number theory unsolved , number theory

Prove that there exist infinitely many pairs $ (a, b)$ of natural numbers not equal to $ 1$ such that $ b^b \plus{}a$ is divisible by $ a^a \plus{}b$.

1994 Austrian-Polish Competition, 5

Tags: number theory unsolved , number theory

Solve in integers the following equation $\frac{1}{2}(x + y)(y + z)(z + x) + (x + y + z)^3 = 1 - xyz$.

2008 Romania Team Selection Test, 5

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

Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]

2013 Mexico National Olympiad, 5

Tags: inequalities , number theory unsolved , number theory

A pair of integers is special if it is of the form $(n, n-1)$ or $(n-1, n)$ for some positive integer $n$. Let $n$ and $m$ be positive integers such that pair $(n, m)$ is not special. Show $(n, m)$ can be expressed as a sum of two or more different special pairs if and only if $n$ and $m$ satisfy the inequality $ n+m\geq (n-m)^2 $. Note: The sum of two pairs is defined as $ (a, b)+(c, d) = (a+c, b+d) $.

1984 IMO Longlists, 15

Tags: number theory unsolved , number theory

Consider all the sums of the form \[\displaystyle\sum_{k=1}^{1985} e_kk^5=\pm 1^5\pm 2^5\pm\cdots\pm1985^5\] where $e_k=\pm 1$. What is the smallest nonnegative value attained by a sum of this type?

2014 Iran Team Selection Test, 4

Tags: quadratics , number theory unsolved , number theory

$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube). $(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation. $(b)$ Prove that for no natural number $n$ exists a cubic permutation.

2010 Kurschak Competition, 3

Tags: number theory unsolved , number theory

For what positive integers $n$ and $k$ do there exits integers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_k$ such that the products $a_ib_j$ ($1\le i\le n,1\le j\le k$) give pairwise different residues modulo $nk$?

2012 Kazakhstan National Olympiad, 1

Tags: modular arithmetic , number theory unsolved , number theory , Kazakhstan

The number $\overline{13\ldots 3}$, with $k>1$ digits $3$, is a prime. Prove that $6\mid k^{2}-2k+3$.

2011 Akdeniz University MO, 2

Tags: number theory , number theory unsolved

Let $a$ and $b$ is roots of the $x^2-6x+1$ equation. [b]a[/b]) Show that, for all $n \in{\mathbb Z^+}$ , $a^n+b^n$ is a integer. [b]b[/b]) Show that, for all $n \in{\mathbb Z^+}$ , $5$ isn't divide $a^n+b^n$

2014 ELMO Shortlist, 5

Tags: inequalities , induction , number theory , number theory unsolved

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

1979 IMO Longlists, 55

Tags: number theory , greatest common divisor , search , relatively prime , number theory unsolved

Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.

2004 Baltic Way, 9

Tags: pigeonhole principle , number theory unsolved , number theory

A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.

2005 Postal Coaching, 11

Tags: number theory unsolved , number theory

(a) Prove that the set $X = (1,2,....100)$ cannot be partitoned into THREE subsets such that two numbers differing by a square belong to different subsets. (b) Prove that $X$ can so be partitioned into $5$ subsets.

2002 Mediterranean Mathematics Olympiad, 1

Tags: search , number theory unsolved , number theory

Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.

2003 China Team Selection Test, 2

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

Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

2000 France Team Selection Test, 3

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

Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.

1980 IMO, 8

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

Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).

2002 Italy TST, 2

Tags: modular arithmetic , number theory unsolved , number theory

Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides \[\binom{p^n}{p}-p^{n-1}. \]

2005 India IMO Training Camp, 1

Tags: number theory unsolved , number theory

Let $0 <a <b$ be two rational numbers. Let $M$ be a set of positive real numbers with the properties: (i) $a \in M$ and $b \in M$; (ii) if $x$ $\in M$ and $y \in M$, then $\sqrt{xy} \in M$. Let $M^*$denote the set of all irrational numbers in $M$. prove that every $c,d$ such that $a <c <d<b$, $M^*$ contains an element $m$ with property $c<m<d$

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 $

2005 France Team Selection Test, 1

Tags: number theory unsolved , number theory

Let $x$, $y$ be two positive integers such that $\displaystyle 3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.