Olympiad problems

2010 Middle European Mathematical Olympiad, 4

Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

2009 Iran Team Selection Test, 2

Tags: pigeonhole principle , modular arithmetic , inequalities , number theory , prime numbers , Diophantine equation , number theory proposed

Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite

2004 Iran Team Selection Test, 2

Tags: modular arithmetic , number theory proposed , number theory

Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.

2014 Iran Team Selection Test, 3

Tags: algebra , polynomial , number theory proposed , number theory

prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.

2012 Junior Balkan Team Selection Tests - Moldova, 1

Tags: number theory proposed , number theory

Find a sequence of $ 2012 $ distinct integers bigger than $ 0 $ such that their sum is a perfect square and their product is a perfect cube.

2010 Iran MO (3rd Round), 6

Tags: number theory proposed , number theory , Hi

$g$ and $n$ are natural numbers such that $gcd(g^2-g,n)=1$ and $A=\{g^i|i \in \mathbb N\}$ and $B=\{x\equiv (n)|x\in A\}$(by $x\equiv (n)$ we mean a number from the set $\{0,1,...,n-1\}$ which is congruent with $x$ modulo $n$). if for $0\le i\le g-1$ $a_i=|[\frac{ni}{g},\frac{n(i+1)}{g})\cap B|$ prove that $g-1|\sum_{i=0}^{g-1}ia_i$.( the symbol $|$ $|$ means the number of elements of the set)($\frac{100}{6}$ points) the exam time was 4 hours

2007 District Olympiad, 3

Tags: number theory proposed , number theory

Eight consecutive positive integers are divided into 2 sets, such that the sum of the squares of the elements in the first set is equal to the sum of the squares of the elements in the second set. Prove that the sum of the lements in the first set is equal to the sum of the elements in the second one.

2009 ISI B.Stat Entrance Exam, 10

Tags: number theory proposed , number theory

Let $x_n$ be the $n$-th non-square positive integer. Thus $x_1=2, x_2=3, x_3=5, x_4=6,$ etc. For a positive real number $x$, denotes the integer closest to it by $\langle x\rangle$. If $x=m+0.5$, where $m$ is an integer, then define $\langle x\rangle=m$. For example, $\langle 1.2\rangle =1, \langle 2.8 \rangle =3, \langle 3.5\rangle =3$. Show that $x_n=n+\langle \sqrt{n}\rangle$

2014 USA Team Selection Test, 2

Tags: quadratics , USA(J)MO , USAMO , induction , modular arithmetic , number theory proposed , Quadratic Residues

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

2001 Hong kong National Olympiad, 2

Tags: inequalities , number theory proposed , number theory

Find, with proof, all positive integers $n$ such that the equation $x^{3}+y^{3}+z^{3}=nx^{2}y^{2}z^{2}$ has a solution in positive integers.

2003 Flanders Math Olympiad, 3

Tags: number theory proposed , number theory

A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.

2009 India IMO Training Camp, 8

Tags: modular arithmetic , number theory proposed , number theory

Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.

2014 Silk Road, 4

Tags: induction , inequalities , number theory proposed , number theory

Find all $ f:N\rightarrow N$, such that $\forall m,n\in N $ $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $

2006 QEDMO 3rd, 5

Tags: Pascal's Triangle , number theory proposed , number theory

Find all positive integers $n$ such that there are $\infty$ many lines of Pascal's triangle that have entries coprime to $n$ only. In other words: such that there are $\infty$ many $k$ with the property that the numbers $\binom{k}{0},\binom{k}{1},\binom{k}{2},...,\binom{k}{k}$ are all coprime to $n$.

2011 Turkey Team Selection Test, 3

Tags: function , modular arithmetic , induction , number theory proposed , number theory

Let $p$ be a prime, $n$ be a positive integer, and let $\mathbb{Z}_{p^n}$ denote the set of congruence classes modulo $p^n.$ Determine the number of functions $f: \mathbb{Z}_{p^n} \to \mathbb{Z}_{p^n}$ satisfying the condition \[ f(a)+f(b) \equiv f(a+b+pab) \pmod{p^n} \] for all $a,b \in \mathbb{Z}_{p^n}.$

2002 Baltic Way, 5

Tags: number theory proposed , number theory

Find all pairs $(a,b)$ of positive rational numbers such that \[\sqrt{a}+\sqrt{b}=\sqrt{2+\sqrt{3}}. \]

2001 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory proposed , number theory

Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!

2001 Turkey Team Selection Test, 1

Tags: number theory proposed , number theory

Find all ordered pairs of integers $(x,y)$ such that $5^x = 1 + 4y + y^4$.

2012 Iran MO (3rd Round), 1

Tags: algebra , polynomial , modular arithmetic , inequalities , number theory , prime numbers , number theory proposed

$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$. [i]Proposed by Mohammad Gharakhani[/i]

1995 Balkan MO, 3

Tags: modular arithmetic , number theory proposed , number theory , Fermat s Little Theorem

Let $a$ and $b$ be natural numbers with $a > b$ and having the same parity. Prove that the solutions of the equation \[ x^2 - (a^2 - a + 1)(x - b^2 - 1) - (b^2 + 1)^2 = 0 \] are natural numbers, none of which is a perfect square. [i]Albania[/i]

2013 International Zhautykov Olympiad, 2

Tags: modular arithmetic , number theory proposed , number theory

Find all odd positive integers $n>1$ such that there is a permutation $a_1, a_2, a_3, \ldots, a_n$ of the numbers $1, 2,3, \ldots, n$ where $n$ divides one of the numbers $a_k^2 - a_{k+1} - 1$ and $a_k^2 - a_{k+1} + 1$ for each $k$, $1 \leq k \leq n$ (we assume $a_{n+1}=a_1$).

2000 Italy TST, 1

Tags: number theory proposed , number theory

Determine all triples $(x,y,z)$ of positive integers such that \[\frac{13}{x^2}+\frac{1996}{y^2}=\frac{z}{1997} \]

2000 Iran MO (2nd round), 1

Tags: number theory proposed , number theory , algebra , set theory , Subsets

Find all positive integers $n$ such that we can divide the set $\{1,2,3,\ldots,n\}$ into three sets with the same sum of members.

2010 All-Russian Olympiad, 3

Tags: number theory proposed , number theory

Let us call a natural number $unlucky$ if it cannot be expressed as $\frac{x^2-1}{y^2-1} $ with natural numbers $x,y >1$. Is the number of $unlucky$ numbers finite or infinite?

2014 Canada National Olympiad, 3

Tags: number theory proposed , number theory

Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if [list] [*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and [*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and [*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list] Determine the number of good $p$-tuples.