Problem

Source: Kosovo Mathematical Olympiad 2022, Grade 12, Problem 2

Tags: combinatorics



Let be given $n$ positive integer. Lets write with $a_n$ the number of positive integer pairs $(x,y)$ such that $x+y$ is even and $1\leq x\leq y\leq n$. Lets write with $b_n$ the number of positive integer pairs $(x,y)$ such that $x+y\leq n+1$ and $1\leq x\leq y\leq n$.