Olympiad problems

2008 Greece JBMO TST, 3

Let $x_1,x_2,x_3,...,x_{102}$ be natural numbers such that $x_1<x_2<x_3<...<x_{102}<255$. Prove that among the numbers $d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}$ there are at least $26$ equal.

2007 Singapore Junior Math Olympiad, 4

Tags: number theory , greatest common divisor , least common multiple , Sum , Difference , Product

The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.

2002 Cono Sur Olympiad, 5

Tags: combinatorics , number theory , maximum , Difference , cono sur

Consider the set $A = \{1, 2, ..., n\}$. For each integer $k$, let $r_k$ be the largest quantity of different elements of $A$ that we can choose so that the difference between two numbers chosen is always different from $k$. Determine the highest value possible of $r_k$, where $1 \le k \le \frac{n}{2}$

1994 All-Russian Olympiad Regional Round, 10.5

Tags: number theory , primes , prime , Difference

Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).

2009 Thailand Mathematical Olympiad, 2

Tags: combinatorics , Subsets , Difference

Let $k$ and $n$ be positive integers with $k < n$. Find the number of subsets of $\{1, 2, . . . , n\}$ such that the difference between the largest and smallest elements in the subset is $k$.