Let \( f : [0, 1] \to \mathbb{R} \) be a continuously differentiable function with \( f(0) = 0 \) and \( 0 < f'(x) \leq 1 \). Prove that \[ \left( \int_0^1 f(x) dx \right)^2 \geq \int_0^1 [f(x)]^3 dx. \]

The best solution was submitted by 박훈민, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +3), 서기원(09학번, +3), 김범수(10학번, +3), 황성호(13학번, +3), 홍혁표(13학번, +3), 김준(13학번, +3), 전한솔(고려대 13학번, +3), 이시우(POSTECH 13학번, +3), 한대진(신현여중 교사, +3). Thank you for your participation.