Olympiad problems

2000 Vietnam Team Selection Test, 1

Let $a, b, c$ be pairwise coprime natural numbers. A positive integer $n$ is said to be [i]stubborn[/i] if it cannot be written in the form $n = bcx+cay+abz$, for some $x, y, z \in\mathbb{ N}.$ Determine the number of stubborn numbers.

2008 District Olympiad, 3

Tags: floor function , number theory proposed , number theory

Prove that if $ n\geq 4$, $ n\in\mathbb Z$ and $ \left \lfloor \frac {2^n}{n} \right\rfloor$ is a power of 2, then $ n$ is also a power of 2.

2011 Baltic Way, 20

Tags: Diophantine equation , number theory proposed , number theory

An integer $n\ge 1$ is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers $n$ such that there are exactly two balanced numbers among $n,n+1,n+2$ and $n+3$.

2012 Serbia National Math Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all natural numbers $n$ for which there is a permutation $(p_1,p_2,...,p_n)$ of numbers $(1,2,...,n)$ such that sets $\{p_1 +1, p_2 + 2,..., p_n +n\}$ and $\{p_1-1, p_2-2,...,p_n -n\}$ are complete residue systems $\mod n$.

2018 Iran Team Selection Test, 5

Tags: number theory proposed

Prove that for each positive integer $m$, one can find $m$ consecutive positive integers like $n$ such that the following phrase doesn't be a perfect power: $$\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)$$ [i]Proposed by Navid Safaei[/i]

2012 India IMO Training Camp, 2

Tags: number theory proposed , number theory

Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.

2012 Greece National Olympiad, 1

Tags: parameterization , number theory proposed , number theory

Let positive integers $p,q$ with $\gcd(p,q)=1$ such as $p+q^2=(n^2+1)p^2+q$. If the parameter $n$ is a positive integer, find all possible couples $(p,q)$.

2007 District Olympiad, 4

Tags: number theory proposed , number theory

Let $n$ be a positive integer which is not prime. Prove that there exist $k, a_{1},a_{2},...a_{k}>1$ positive integers such that $a_{1}+a_{2}+\cdots+a_{k}=n(\frac1{a_{1}}+\frac1{a_{2}}+\cdots+\frac1{a_{k}})$ Edit: the $a_{i}'s$ have to be grater than 1. Sorry, my mistake :blush:

2006 India IMO Training Camp, 1

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

Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.

1995 Baltic Way, 3

Tags: number theory , relatively prime , number theory proposed

The positive integers $a,b,c$ are pairwise relatively prime, $a$ and $c$ are odd and the numbers satisfy the equation $a^2+b^2=c^2$. Prove that $b+c$ is the square of an integer.

2003 Federal Math Competition of S&M, Problem 4

Tags: number theory proposed , number theory

Let $S$ be the subset of $N$($N$ is the set of all natural numbers) satisfying: i)Among each $2003$ consecutive natural numbers there exist at least one contained in $S$; ii)If $n \in S$ and $n>1$ then $[\frac{n}{2}] \in S$ Prove that:$S=N$ I hope it hasn't posted before. :lol: :lol:

2001 Taiwan National Olympiad, 1

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

Let $A$ be a set with at least $3$ integers, and let $M$ be the maximum element in $A$ and $m$ the minimum element in $A$. it is known that there exist a polynomial $P$ such that: $m<P(a)<M$ for all $a$ in $A$. And also $p(m)<p(a)$ for all $a$ in $A-(m,M)$. Prove that $n<6$ and there exist integers $b$ and $c$ such that $p(x)+x^2+bx+c$ is cero in $A$.

2018 Junior Balkan Team Selection Tests - Romania, 1

Tags: floor function , number theory , number theory proposed

Let $ p$ be a prime number with $ p>5$. Consider the set $ X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$. Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$. [i]Albania[/i]

1990 IMO Longlists, 57

Tags: induction , algebra , polynomial , number theory , greatest common divisor , number theory proposed

The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$

2000 Iran MO (3rd Round), 1

Tags: induction , number theory proposed , number theory

A sequence of natural numbers $c_1, c_2,\dots$ is called [i]perfect[/i] if every natural number $m$ with $1\le m \le c_1 +\dots+ c_n$ can be represented as $m =\frac{c_1}{a_1}+\frac{c_2}{a_2}+\dots+\frac{c_n}{a_n}$ Given $n$, find the maximum possible value of $c_n$ in a perfect sequence $(c_i)$.

2017 Bulgaria JBMO TST, 4

Tags: number theory , number theory proposed

Find all positive integers such that they have $6$ divisors (without $1$ and the number itself) and the sum of the divisors is $14133$.

2010 Slovenia National Olympiad, 1

Tags: modular arithmetic , number theory , number theory proposed

Let $a,b,c$ be positive integers. Prove that $a^2+b^2+c^2$ is divisible by $4$, if and only if $a,b,c$ are even.

2008 China Western Mathematical Olympiad, 3

Tags: number theory , number theory proposed

Given an integer $ m\geq$ 2, m positive integers $ a_1,a_2,...a_m$. Prove that there exist infinitely many positive integers n, such that $ a_{1}1^{n} \plus{} a_{2}2^{n} \plus{} ... \plus{} a_{m}m^{n}$ is composite.

2001 Romania National Olympiad, 4

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.

2006 Australia National Olympiad, 2

Tags: number theory proposed , number theory

For any positive integer $n$, define $a_n$ to be the product of the digits of $n$. (a) Prove that $n \geq a(n)$ for all positive integers $n$. (b) Find all $n$ for which $n^2-17n+56 = a(n)$.

2010 Hong kong National Olympiad, 3

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

Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that \[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\] where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.

2013 Middle European Mathematical Olympiad, 4

Tags: number theory proposed , number theory

Let $ a$ and $b$ be positive integers. Prove that there exist positive integers $ x $ and $ y $ such that \[ \binom{x+y}{2} = ax + by . \]

1974 IMO Longlists, 2

Tags: number theory , prime numbers , number theory proposed

Let ${u_n}$ be the Fibonacci sequence, i.e., $u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}$ for $n>1$. Prove that there exist infinitely many prime numbers $p$ that divide $u_{p-1}$.

1997 Baltic Way, 7

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

Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.

1999 CentroAmerican, 2

Tags: modular arithmetic , number theory proposed , number theory

Find a positive integer $n$ with 1000 digits, all distinct from zero, with the following property: it's possible to group the digits of $n$ into 500 pairs in such a way that if the two digits of each pair are multiplied and then add the 500 products, it results a number $m$ that is a divisor of $n$.