CMO 2010 Problem 1

For a positive integer $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{\text {th }}$ row, such that all the left-most squares in each row are aligned vertically. For example, the 5 -staircase is shown below.

Let $f(n)$ denote the minimum number of square tiles required to tile the $n$ staircase, where the side lengths of the square tiles can be any positive integer. For example, $f(2)=3$ and $f(4)=7$.

(a) Find all $n$ such that $f(n)=n$.

(b) Find all $n$ such that $f(n)=n+1$.