Olympiad problems

2000 Balkan MO, 4

Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.

2005 Baltic Way, 17

Tags: number theory proposed , number theory , algebraic identity

A sequence $(x_n)_{n\ge 0}$ is defined as follows: $x_0=a,x_1=2$ and $x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1$ for all $n>1$. Find all integers $a$ such that $2x_{3n}-1$ is a perfect square for all $n\ge 1$.

1998 Balkan MO, 1

Tags: floor function , number theory proposed , number theory

Consider the finite sequence $\left\lfloor \frac{k^2}{1998} \right\rfloor$, for $k=1,2,\ldots, 1997$. How many distinct terms are there in this sequence? [i]Greece[/i]

1997 Turkey Team Selection Test, 2

Tags: modular arithmetic , abstract algebra , number theory proposed , number theory

Show that for each prime $p \geq 7$, there exist a positive integer $n$ and integers $x_{i}$, $y_{i}$ $(i = 1, . . . , n)$, not divisible by $p$, such that $x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}$ where $x_{n+1} = x_{1}$

2010 Junior Balkan MO, 2

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

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

2004 Iran Team Selection Test, 1

Tags: modular arithmetic , quadratics , special factorizations , number theory proposed , number theory

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

2005 Indonesia MO, 2

Tags: number theory proposed , number theory

For an arbitrary positive integer $ n$, define $ p(n)$ as the product of the digits of $ n$ (in decimal). Find all positive integers $ n$ such that $ 11p(n)\equal{}n^2\minus{}2005$.

2001 Romania National Olympiad, 2

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

For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers. a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty. b) Show that if $G_p\cap G_q$ is a point with integer coordinates, then $p$ and $q$ are integer numbers. c) Show that if $p,q,r$ are consecutive natural numbers, then the area of the triangle determined by intersections of $G_p,G_q$ and $G_r$ is equal to $1$.

2006 Romania Team Selection Test, 2

Tags: quadratics , real analysis , complex numbers , algebra , polynomial , sum of roots , number theory proposed

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2004 Regional Olympiad - Republic of Srpska, 3

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

Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite. If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?

1988 Romania Team Selection Test, 9

Tags: induction , number theory proposed , number theory

Prove that for all positive integers $n\geq 1$ the number $\prod^n_{k=1} k^{2k-n-1}$ is also an integer number. [i]Laurentiu Panaitopol[/i].

2010 ELMO Shortlist, 1

Tags: inequalities , function , number theory proposed , number theory

For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have \[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \] and determine when equality holds. [i]Wenyu Cao.[/i]

2006 Iran Team Selection Test, 1

Tags: number theory , least common multiple , function , number theory proposed

Suppose that $p$ is a prime number. Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have \[ n|a^{\frac{\varphi(n)}{p}}-1 \]

2012 Regional Competition For Advanced Students, 2

Tags: number theory proposed , number theory

Determine all integer solutions $(x, y)$ of the equation \[(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}\]

2010 ELMO Shortlist, 1

Tags: inequalities , function , number theory proposed , number theory

For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have \[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \] and determine when equality holds. [i]Wenyu Cao.[/i]

1997 All-Russian Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$. [i]M. Sonkin[/i]

2002 China Western Mathematical Olympiad, 2

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

Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions: $ (1): a_{1}\plus{}a_{2}\plus{}\cdots\plus{}a_{n}\ge n^2;$ $ (2): a_{1}^2\plus{}a_{2}^2\plus{}\cdots\plus{}a_{n}^2\le n^3\plus{}1.$

2013 ELMO Shortlist, 4

Tags: number theory , number theory proposed , Hi

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

2022 Turkey Team Selection Test, 1

Tags: number theory , modular arithmetic , number theory proposed , Diophantine equation

Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

2005 Junior Balkan Team Selection Tests - Romania, 12

Tags: modular arithmetic , number theory proposed , number theory

Find all positive integers $n$ and $p$ if $p$ is prime and \[ n^8 - p^5 = n^2+p^2 . \] [i]Adrian Stoica[/i]

1992 IberoAmerican, 1

Tags: number theory proposed , number theory

For every positive integer $n$ we define $a_{n}$ as the last digit of the sum $1+2+\cdots+n$. Compute $a_{1}+a_{2}+\cdots+a_{1992}$.

2010 Regional Competition For Advanced Students, 4

Tags: number theory proposed , number theory

Let $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$ for positive integers $a_0$ and $d$. We consider all such sequences containing an element $b_i$ which equals $2010$. Determine the greatest possible value of $i$ and for this value the integers $a_0$ and $d$. [i](41th Austrian Mathematical Olympiad, regional competition, problem 4)[/i]

1992 Hungary-Israel Binational, 1

Tags: modular arithmetic , number theory proposed , number theory

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. Prove that $1+L_{2^{j}}\equiv 0 \pmod{2^{j+1}}$ for $j \geq 0$.

2013 Korea National Olympiad, 5

Tags: number theory , greatest common divisor , least common multiple , function , number theory proposed

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying \[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \] for all positive integer $m,n$.

2009 Tournament Of Towns, 4

Tags: factorial , function , floor function , number theory proposed , number theory

Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$ [i](8 points)[/i]