Olympiad problems

2011 Tournament of Towns, 1

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.

2007 Singapore Junior Math Olympiad, 4

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.

least common multiple greatest common divisor number theory sum product difference

1975 Chisinau City MO, 87

Prove that among any $100$ natural numbers there are two numbers whose difference is divisible by $99$.

divisible number theory difference

2015 JBMO Shortlist, NT3

a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$ b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$ PS. a) original from Albania b) modified by problem selecting committee

divisible number theory product difference parity pigeonhole principle

2025 Philippine MO, P1

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?

set combinatorics subset difference