Olympiad problems

1996 Balkan MO, 2

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]

2007 Middle European Mathematical Olympiad, 4

Tags: induction , number theory proposed , number theory

Determine all pairs $ (x,y)$ of positive integers satisfying the equation \[ x!\plus{}y!\equal{}x^{y}.\]

2014 Kazakhstan National Olympiad, 3

Tags: inequalities , number theory proposed , number theory , Kazakhstan

Prove that, for all $n\in\mathbb{N}$, on $ [n-4\sqrt{n}, n+4\sqrt{n}]$ exists natural number $k=x^3+y^3$ where $x$, $y$ are nonnegative integers.

2013 IberoAmerican, 3

Tags: induction , number theory proposed , number theory , Iberoamerican

Let $A = \{1,...,n\}$ with $n \textgreater 5$. Prove that one can find $B$ a finite set of positive integers such that $A$ is a subset of $B$ and $\displaystyle\sum_{x \in B} x^2 = \displaystyle\prod_{x \in B} x$

1999 Taiwan National Olympiad, 1

Tags: number theory proposed , number theory

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

1986 IMO Longlists, 4

Tags: number theory proposed , number theory

Find the last eight digits of the binary development of $27^{1986}.$

2007 China Team Selection Test, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \plus{} b \minus{} 1$ is a power of prime number $ ; a^2 \plus{} b \plus{} 1$ can divide $ b^2 \minus{} a^3 \minus{} 1,$ but it can't divide $ (a \plus{} b \minus{} 1)^2.$

2003 Baltic Way, 20

Tags: modular arithmetic , number theory proposed , number theory

Suppose that the sum of all positive divisors of a natural number $n$, $n$ excluded, plus the number of these divisors is equal to $n$. Prove that $n = 2m^2$ for some integer $m$.

1994 Baltic Way, 9

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

Find all pairs of positive integers $(a,b)$ such that $2^a+3^b$ is the square of an integer.

2007 Baltic Way, 20

Tags: geometry , search , function , number theory , number theory proposed

Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.

2011 ELMO Shortlist, 1

Tags: modular arithmetic , number theory proposed , number theory

Prove that $n^3-n-3$ is not a perfect square for any integer $n$. [i]Calvin Deng.[/i]

2012 European Mathematical Cup, 1

Tags: number theory , prime numbers , number theory proposed

Find all positive integers $a$, $b$, $n$ and prime numbers $p$ that satisfy \[ a^{2013} + b^{2013} = p^n\text{.}\] [i]Proposed by Matija Bucić.[/i]

2009 Middle European Mathematical Olympiad, 11

Tags: number theory proposed , number theory

Find all pairs $ (m$, $ n)$ of integers which satisfy the equation \[ (m \plus{} n)^4 \equal{} m^2n^2 \plus{} m^2 \plus{} n^2 \plus{} 6mn.\]

2009 Turkey MO (2nd round), 3

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

If $1<k_1<k_2<...<k_n$ and $a_1,a_2,...,a_n$ are integers such that for every integer $N,$ $k_i \mid N-a_i$ for some $1 \leq i \leq n,$ find the smallest possible value of $n.$

2016 Korea Winter Program Practice Test, 4

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

$p(x)$ is an irreducible polynomial with integer coefficients, and $q$ is a fixed prime number. Let $a_n$ be a number of solutions of the equation $p(x)\equiv 0\mod q^n$. Prove that we can find $M$ such that $\{a_n\}_{n\ge M}$ is constant.

2008 Bulgaria Team Selection Test, 1

Tags: number theory proposed , number theory

For each positive integer $n$, denote by $a_{n}$ the first digit of $2^{n}$ (base ten). Is the number $0.a_{1}a_{2}a_{3}\cdots$ rational?

2014 China Team Selection Test, 2

Tags: function , induction , number theory , relatively prime , number theory proposed

Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying: (1)$\tau (n)=a$ (2)$n|\phi (n)+\sigma (n)$ Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.

2000 Taiwan National Olympiad, 1

Tags: inequalities , number theory proposed , number theory

Find all pairs $(x,y)$ of positive integers such that $y^{x^2}=x^{y+2}$.

2003 India IMO Training Camp, 7

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

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

2019 Romania Team Selection Test, 2

Tags: algebra , polynomial , number theory , greatest common divisor , modular arithmetic , number theory proposed , nice problem

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

2006 Iran MO (2nd round), 1

Tags: number theory proposed , number theory

[b]a.)[/b] Let $m>1$ be a positive integer. Prove there exist finite number of positive integers $n$ such that $m+n|mn+1$. [b]b.)[/b] For positive integers $m,n>2$, prove that there exists a sequence $a_0,a_1,\cdots,a_k$ from positive integers greater than $2$ that $a_0=m$, $a_k=n$ and $a_i+a_{i+1}|a_ia_{i+1}+1$ for $i=0,1,\cdots,k-1$.

1998 Spain Mathematical Olympiad, 2

Tags: geometry , 3D geometry , modular arithmetic , number theory proposed , number theory

Find all four-digit numbers which are equal to the cube of the sum of their digits.

2005 Vietnam National Olympiad, 2

Tags: Ross Mathematics Program , number theory proposed , number theory

Find all triples of natural $ (x,y,n)$ satisfying the condition: \[ \frac {x! \plus{} y!}{n!} \equal{} 3^n \] Define $ 0! \equal{} 1$

2008 China Team Selection Test, 3

Tags: analytic geometry , geometry , modular arithmetic , trapezoid , induction , trigonometry , number theory proposed

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

2011 ELMO Shortlist, 3

Tags: number theory proposed , number theory

Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ [i]good[/i] if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples. [i]Mitchell Lee.[/i]