Olympiad problems

2006 All-Russian Olympiad, 2

Show that there exist four integers $a$, $b$, $c$, $d$ whose absolute values are all $>1000000$ and which satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{abcd}$.

2011 Baltic Way, 19

Tags: arithmetic sequence , number theory proposed , number theory

Let $p\neq 3$ be a prime number. Show that there is a non-constant arithmetic sequence of positive integers $x_1,x_2,\ldots ,x_p$ such that the product of the terms of the sequence is a cube.

2007 Moldova Team Selection Test, 4

Tags: number theory , prime numbers , number theory proposed

Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.

2005 Germany Team Selection Test, 3

Tags: induction , number theory proposed , number theory

Let $b$ and $c$ be any two positive integers. Define an integer sequence $a_n$, for $n\geq 1$, by $a_1=1$, $a_2=1$, $a_3=b$ and $a_{n+3}=ba_{n+2}a_{n+1}+ca_n$. Find all positive integers $r$ for which there exists a positive integer $n$ such that the number $a_n$ is divisible by $r$.

2010 India IMO Training Camp, 12

Tags: number theory proposed , number theory

Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$

2012 India IMO Training Camp, 2

Tags: modular arithmetic , inequalities , number theory proposed , number theory

Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent: $(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$ $(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$

2016 India National Olympiad, P6

Tags: arithmetic sequence , number theory , relatively prime , number theory proposed

Consider a nonconstant arithmetic progression $a_1, a_2,\cdots, a_n,\cdots$. Suppose there exist relatively prime positive integers $p>1$ and $q>1$ such that $a_1^2, a_{p+1}^2$ and $a_{q+1}^2$ are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.

2014 Moldova Team Selection Test, 1

Tags: quadratics , number theory proposed , number theory

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

1997 Baltic Way, 17

Tags: geometry , rectangle , ratio , number theory proposed , number theory

A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$.

2006 Vietnam Team Selection Test, 2

Tags: number theory proposed , number theory

Find all pair of integer numbers $(n,k)$ such that $n$ is not negative and $k$ is greater than $1$, and satisfying that the number: \[ A=17^{2006n}+4.17^{2n}+7.19^{5n} \] can be represented as the product of $k$ consecutive positive integers.

2006 IMS, 2

Tags: induction , number theory proposed , number theory

For each subset $C$ of $\mathbb N$, Suppose $C\oplus C=\{x+y|x,y\in C, x\neq y\}$. Prove that there exist a unique partition of $\mathbb N$ to sets $A$, $B$ that $A\oplus A$ and $B\oplus B$ do not have any prime numbers.

2007 Iran MO (2nd Round), 1

Tags: number theory proposed , number theory

Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.

2011 Romanian Masters In Mathematics, 1

Tags: function , number theory proposed , number theory

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$). Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$; ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$. [i](Romania) Dan Schwarz[/i]

1996 Turkey MO (2nd round), 2

Tags: group theory , abstract algebra , floor function , modular arithmetic , number theory proposed , number theory

Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.

2010 China Second Round Olympiad, 2

Tags: modular arithmetic , number theory proposed , number theory

Given a fixed integer $k>0,r=k+0.5$,define $f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$ where $[x]$ denotes the smallest integer not less than $x$. prove that there exists integer $m$ such that $f^m(r)$ is an integer.

2001 Baltic Way, 19

Tags: number theory proposed , number theory

What is the smallest positive odd integer having the same number of positive divisors as $360$?

2010 Iran MO (3rd Round), 2

Tags: geometry , 3D geometry , number theory , relatively prime , number theory proposed

[b]rolling cube[/b] $a$,$b$ and $c$ are natural numbers. we have a $(2a+1)\times (2b+1)\times (2c+1)$ cube. this cube is on an infinite plane with unit squares. you call roll the cube to every side you want. faces of the cube are divided to unit squares and the square in the middle of each face is coloured (it means that if this square goes on a square of the plane, then that square will be coloured.) prove that if any two of lengths of sides of the cube are relatively prime, then we can colour every square in plane. time allowed for this question was 1 hour.

2010 IFYM, Sozopol, 1

Tags: number theory proposed , number theory

Determine the ordered systems $(x,y,z)$ of positive rational numbers for which $x+\frac{1}{y},y+\frac{1}{z}$ and $z+\frac{1}{x}$ are integers.

1988 India National Olympiad, 1

Tags: number theory proposed , number theory

Let $ m_1,m_2,m_3,\dots,m_n$ be a rearrangement of the numbers $ 1,2,\dots,n$. Suppose that $ n$ is odd. Prove that the product \[ \left(m_1\minus{}1\right)\left(m_2\minus{}2\right)\cdots \left(m_n\minus{}n\right)\] is an even integer.

2010 All-Russian Olympiad, 3

Tags: logarithms , number theory , prime numbers , number theory proposed

Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $.

2003 Iran MO (3rd Round), 22

Tags: induction , number theory proposed , number theory

Let $ a_1\equal{}a_2\equal{}1$ and \[ a_{n\plus{}2}\equal{}\frac{n(n\plus{}1)a_{n\plus{}1}\plus{}n^2a_n\plus{}5}{n\plus{}2}\minus{}2\]for each $ n\in\mathbb N$. Find all $ n$ such that $ a_n\in\mathbb N$.

1971 IMO Longlists, 53

Tags: limit , floor function , logarithms , number theory proposed , number theory

Denote by $x_n(p)$ the multiplicity of the prime $p$ in the canonical representation of the number $n!$ as a product of primes. Prove that $\frac{x_n(p)}{n}<\frac{1}{p-1}$ and $\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}$.

1990 IberoAmerican, 3

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

Let $b$, $c$ be integer numbers, and define $f(x)=(x+b)^2-c$. i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$, show that for every integer $n$, $f(n)$ is not divisible by $p^{2}$. ii) Let $q \neq 2$ be a prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$.

2006 Iran MO (3rd Round), 6

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

a) $P(x),R(x)$ are polynomials with rational coefficients and $P(x)$ is not the zero polynomial. Prove that there exist a non-zero polynomial $Q(x)\in\mathbb Q[x]$ that \[P(x)\mid Q(R(x)).\] b) $P,R$ are polynomial with integer coefficients and $P$ is monic. Prove that there exist a monic polynomial $Q(x)\in\mathbb Z[x]$ that \[P(x)\mid Q(R(x)).\]

2013 Albania Team Selection Test, 1

Tags: ratio , function , number theory proposed , number theory

Find the 3-digit number whose ratio with the sum of its digits it's minimal.