Olympiad problems

2011 IFYM, Sozopol, 7

solve $x^2+31=y^3$ in integers

2002 Italy TST, 3

Tags: algebra , polynomial , Vieta , number theory , relatively prime , number theory unsolved

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

2002 Czech-Polish-Slovak Match, 4

Tags: number theory unsolved , number theory

An integer $n > 1$ and a prime $p$ are such that $n$ divides $p-1$, and $p$ divides $n^3 - 1$. Prove that $4p - 3$ is a perfect square.

2004 Bulgaria National Olympiad, 5

Tags: geometry , rhombus , number theory unsolved , number theory

Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$.

1988 IMO Longlists, 45

Tags: quadratics , number theory unsolved , number theory

Let $g(n)$ be defined as follows: \[ g(1) = 0, g(2) = 1 \] and \[ g(n+2) = g(n) + g(n+1) + 1, n \geq 1. \] Prove that if $n > 5$ is a prime, then $n$ divides $g(n) \cdot (g(n) + 1).$

2001 India IMO Training Camp, 1

Tags: algebra , polynomial , number theory unsolved , number theory

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

2009 Korea - Final Round, 6

Tags: modular arithmetic , number theory unsolved , number theory

Find all pairs of two positive integers $(m,n)$ satisfying $ 3^m - 7^n = 2 $.

2010 China Team Selection Test, 3

Tags: modular arithmetic , number theory unsolved , number theory

Fine all positive integers $m,n\geq 2$, such that (1) $m+1$ is a prime number of type $4k-1$; (2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that \[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]

2014 South africa National Olympiad, 4

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

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also \[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\] (b) Show that there are no two positive integers $a$ and $b$ such that \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]

2014 Bosnia Herzegovina Team Selection Test, 3

Tags: modular arithmetic , number theory unsolved , Zsigmondy

Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$

2011 India IMO Training Camp, 1

Tags: geometry , 3D geometry , AMC , AIME , Diophantine equation , number theory unsolved , number theory

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

1989 IMO Longlists, 9

Tags: floor function , number theory unsolved , number theory

Let $ m$ be a positive integer and define $ f(m)$ to be the number of factors of $ 2$ in $ m!$ (that is, the greatest positive integer $ k$ such that $ 2^k|m!$). Prove that there are infinitely many positive integers $ m$ such that $ m \minus{} f(m) \equal{} 1989.$

2010 Indonesia TST, 4

Tags: number theory unsolved , number theory

How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $

2002 Turkey MO (2nd round), 1

Tags: modular arithmetic , quadratics , number theory , prime numbers , number theory unsolved

Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition \[y^2 \equiv x^3 - x \pmod p\] is exactly $p.$

2008 Baltic Way, 8

Tags: number theory unsolved , number theory

Consider a set $ A$ of positive integers such that the least element of $ A$ equals $ 1001$ and the product of all elements of $ A$ is a perfect square. What is the least possible value of the greatest element of $ A$?

2011 Postal Coaching, 6

Tags: number theory unsolved , number theory

A positive integer is called [i]monotonic[/i] if when written in base $10$, the digits are weakly increasing. Thus $12226778$ is monotonic. Note that a positive integer cannot have ﬁrst digit $0$. Prove that for every positive integer $n$, there is an $n$-digit monotonic number which is a perfect square.

2009 Germany Team Selection Test, 1

Tags: number theory , prime numbers , number theory unsolved

For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

2000 Brazil Team Selection Test, Problem 4

Tags: inequalities , number theory , relatively prime , number theory unsolved

[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)

1999 APMO, 1

Tags: arithmetic sequence , number theory unsolved , number theory , Hi

Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers.

2010 South East Mathematical Olympiad, 1

Tags: quadratics , algebra , number theory unsolved , number theory

Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.

2002 Federal Math Competition of S&M, Problem 4

Tags: number theory unsolved , number theory

Is there a positive integer $ k$ such that none of the digits $ 3,4,5,6$ appear in the decimal representation of the number $ 2002!\cdot k$?

1991 Irish Math Olympiad, 1

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

Problem. The sum of two consecutive squares can be a square; for instance $3^2 + 4^2 = 5^2$. (a) Prove that the sum of $m$ consecutive squares cannot be a square for $m \in \{3, 4, 5, 6\}$. (b) Find an example of eleven consecutive squares whose sum is a square. Can anyone help me with this? Thanks.

2005 Hungary-Israel Binational, 3

Tags: number theory unsolved , number theory

Find all sequences $x_{1},x_{2},...,x_{n}$ of distinct positive integers such that $\frac{1}{2}=\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}$.

2002 Olympic Revenge, 7

Tags: olympic revenge 2002 , number theory unsolved , factorial , number theory

Show that \[A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}\] is an integer, for any positive integer $n$.

2010 Contests, 1

Tags: number theory unsolved , number theory

Each of six fruit baskets contains pears, plums and apples. The number of plums in each basket equals the total number of apples in all other baskets combined while the number of apples in each basket equals the total number of pears in all other baskets combined. Prove that the total number of fruits is a multiple of $31$.