Consider in a plane
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
the points
![O,A_1,A_2,A_3,A_4](/media/m/7/9/f/79fdb377ae39d5f5fe493d6e83c80e75.png)
such that
![\sigma(OA_iA_j) \geq 1 \quad \forall i, j = 1, 2, 3, 4, i \neq j.](/media/m/7/b/5/7b512f25d36a938a1dc2c97d06eaff58.png)
where
![\sigma(OA_iA_j)](/media/m/3/a/9/3a949b6c9b4ad2bb01cb7ee703e7e4e9.png)
is the area of triangle
![OA_iA_j.](/media/m/e/d/0/ed02fc79148bfb1c60f7369e82eee67a.png)
Prove that there exists at least one pair
![i_0, j_0 \in \{1, 2, 3, 4\}](/media/m/5/f/d/5fdee5ba744a307696584769230bae64.png)
such that
%V0
Consider in a plane $P$ the points $O,A_1,A_2,A_3,A_4$ such that $$\sigma(OA_iA_j) \geq 1 \quad \forall i, j = 1, 2, 3, 4, i \neq j.$$ where $\sigma(OA_iA_j)$ is the area of triangle $OA_iA_j.$ Prove that there exists at least one pair $i_0, j_0 \in \{1, 2, 3, 4\}$ such that $$\sigma(OA_iA_j) \geq \sqrt{2}.$$