Question 31
Prove that, for all convex pentagons $P_1P_2P_3P_4P_5$ with area $1$, there are indices $i$ and $j$ (assume $P_6 = P_1$ and $P_7 = P_2$) such that:
$$
\text{area } \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt{5}}{10} \le \text{area } \triangle P_j P_{j+1} P_{j+2}
$$
$$
\text{area } \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt{5}}{10} \le \text{area } \triangle P_j P_{j+1} P_{j+2}
$$