Olympiad problems

2002 Cono Sur Olympiad, 5

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 $1 \le k \le \frac{n}{2}$

2025 Philippine MO, P1

Tags: combinatorics , subset , difference , set

The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?

1994 All-Russian Olympiad Regional Round, 10.5

Tags: number theory , 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).

1987 Bundeswettbewerb Mathematik, 4

Tags: 3d geometry , combinatorics , difference , cube , generalization , geometry

Place the integers $1,2 , \ldots, n^{3}$ in the cells of a $n\times n \times n$ cube such that every number appears once. For any possible enumeration, write down the maximal difference between any two adjacent cells (adjacent means having a common vertex). What is the minimal number noted down?

2011 Tournament of Towns, 1

Tags: even , combinatorics , difference

The numbers from $1$ to $2010$ inclusive are placed along a circle so that if we move along the circle in clockwise order, they increase and decrease alternately. Prove that the difference between some two adjacent integers is even.