Putnam 2004 B3

Determine all real numbers $a > 0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region $$R = \{ (x,y) ; 0 \le x \le a, 0 \le y \le f(x) \}$$ has perimeter $k$ units and area $k$ square units for some real number $k$.