OBM 2011 Problem 3

Prove that, for all convex pentagons $P_1 P_2 P_3 P_4 P_5$ with area 1, there are indices $i$ and $j$ (assume $P_7 = P_2$ and $P_6 = P_1$ ) such that:

$$\text{Area of} \ \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt 5}{10} \le \text{Area of} \ \triangle P_j P_{j+1} P_{j+2}$$