Let $e,f$ be the length of the diagonals of the convex quadrangle.We split the problem it to two cases: Case $1$:$x_1$ and $x_2$ are opposite sides.Then $x_3$ and $x_4$ are also opposite sides.By Ptolemey's Inequality it follows that $\frac{1}{2}(x_1x_2+x_3x_4)\ge\frac{1}{2}ef\ge\frac{1}{2}ef\cdot\sin(e,f)=S$.($(e,f)$ means the angle formed between diagonals $e,f$.) In this case we have inequality if and only if the convex quadrangle is cyclic and orthodiagonal. Case $2$:$x_1$ and $x_2$ are not opposite sides.Then $x_3$ and $x_4$ are also not opposite.Then $\frac{1}{2}(x_1x_2+x_3x_4)\ge\frac{1}{2}[x_1x_2\sin(x_1,x_2)+x_3x_4\sin(x_3,x_4)]=S$. In this case we have equality if and only if the angles between $x_1,x_2$ and $x_3,x_4$ are right angles.

Thank you very much.

what about other questions of the competition?, can you post them?

rightways wrote: what about other questions of the competition?, can you post them? CWMI Day 1: Q1 http://www.artofproblemsolving.com/community/c6h1130759 Q2 http://www.artofproblemsolving.com/community/c6t48f6h1132074 Q3 http://www.artofproblemsolving.com/community/c6t243f6h1130397 Q4 http://www.artofproblemsolving.com/community/u222515h1132190 CWMI Day 2: Q5 http://www.artofproblemsolving.com/community/c6t243f6h1130785 Q6 http://www.artofproblemsolving.com/community/u222515h1132199 Q7 http://www.artofproblemsolving.com/community/u222515h1132200 Q8 http://www.artofproblemsolving.com/community/u222515h1132202