Olympiad problems

2021 Science ON all problems, 3

Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$. [i] (From "Radu Păun" contest, Radu Miculescu)[/i]

2021 Science ON grade VI, 3

Tags: Sets , pie , pigeonhole principle , combinatorics

Consider positive integers $a<b$ and the set $C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}$. Suppose $C$ has more than $\frac{b-a+1}{2}$ elements. Prove that there are two elements $x,y\in C$ that satisfy $x+y=a+b$. [i] (From "Radu Păun" contest, Radu Miculescu)[/i]

2022 Indonesia Regional, 1

Tags: Indonesia , RMO , 2022 , pie , combinatorics , number theory

Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has.

2003 JHMMC 8, 20

Tags: JHMMC , pie , Basic PIE

How many positive whole numbers less than $100$ are divisible by $3$, but not by $2$?