International Contests Tuymaada Olympiad 2003 Tuymaada Olympiad 1 (\sum 1/sin(a_i))(\sum 1/cos(a_i)) <= 2*(\sum 1/sin(2a_i))². Problem Statement Prove that for every α 1 , α 2 , … , α n \alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} α 1 , α 2 , … , α n ( 0 , π / 2 ) (0,\pi/2) ( 0 , π /2 ) ( 1 sin α 1 + 1 sin α 2 + … + 1 sin α n ) ( 1 cos α 1 + 1 cos α 2 + … + 1 cos α n ) ≤ \left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq ( sin α 1 1 + sin α 2 1 + … + sin α n 1 ) ( cos α 1 1 + cos α 2 1 + … + cos α n 1 ) ≤ ≤ 2 ( 1 sin 2 α 1 + 1 sin 2 α 2 + … + 1 sin 2 α n ) 2 . \leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}. ≤ 2 ( sin 2 α 1 1 + sin 2 α 2 1 + … + sin 2 α n 1 ) 2 .