Subcontests
(4)Inequality
Prove that among any 42 numbers from the interval [1,106], you can choose four numbers so that for any permutation (a,b,c,d) of these numbers, the inequality
25(ab+cd)(ad+bc)≥16(ac+bd)2
holds. Problem 2, Geometry
The quadrilateral ABCD is inscribed in the circle ω. The diagonals AC and BD intersect at the point O. On the segments AO and DO, the points E and F are chosen, respectively. The straight line EF intersects ω at the points E1 and F1. The circumscribed circles of the triangles ADE and BCF intersect the segment EF at the points E2 and F2 respectively (assume that all the points E,F,E1,F1,E2 and F2 are different). Prove that E1E2=F1F2. (N.Sedrakyan)