HMMT November

Let $a$ be the width of the rectangle. Then the length of the rectangle is $3 a$, so the perimeter is $2(a+3 a)=8 a$, and the area is $3 a^{2}$. Since the length is numerically equal to the width, we know that $$8 a=3 a^{2}=k$$ Because $k>0$, the rectangle is non-degenerate. It follows that $8=3 a$, so $a=\frac{8}{3}$. Therefore, $k=\frac{64}{3}$.