Olympiad problems

2004 China Team Selection Test, 2

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2022 Iran MO (3rd Round), 3

We have many $\text{three-element}$ subsets of a $1000\text{-element}$ set. We know that the union of every $5$ of them has at least $12$ elements. Find the most possible value for the number of these subsets.

union set combinatorics construction graph modeling mathematical induction

2022 IMO, 3

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.

number theory prime number geometry graph modeling mathematical induction modular arithmetic prime numbers

2022 Thailand TSTST, 2

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

greatest common divisor combinatorics graph theory construction graph modeling modular arithmetic number theory proof by contradiction