Question 28
(ROM 2) Consider in a plane $\Pi$ the points $O,A_1,A_2,A_3,A_4$ such that $\sigma(OA_iA_j)\ge1$ for all $i,j=1,2,3,4$, $i\ne j$. Prove that there is at least one pair $i_0,j_0\in\{1,2,3,4\}$ such that $\sigma(OA_{i_0}A_{j_0})\ge\sqrt2$.
(We have denoted by $\sigma(OA_iA_j)$ the area of triangle $OA_iA_j$.)
(We have denoted by $\sigma(OA_iA_j)$ the area of triangle $OA_iA_j$.)