IZhO 2016

$a_1,a_2,...,a_{100}$ are permutation of $1,2,...,100$ . $S_1=a_1, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}$ Find the maximum number of perfect squares from $S_i$

A convex hexagon $ABCDEF$ is given such that $AB||DE$ , $BC||EF$ , $CD||FA$ . The point $M, N, K$ are common points of the lines $BD$ and $AE$ , $AC$ and $DF$ , $CE$ and $BF$ respectively. Prove that perpendiculars drawn from $M, N, K$ to lines $AB, CD, EF$ respectively concurrent.