Let $F=\{1,2,...,100\}$ and let $G$ be any $10$-element subset of $F$. Prove that there exist two disjoint nonempty subsets $S$ and $T$ of $G$ with the same sum of elements.

Suppose that in the exterior of a convex quadrilateral $ABCD$ equilateral triangles $XAB,YBC,ZCD,WDA$ with centroids $S_1,S_2,S_3,S_4$ respectively are constructed. Prove that $S_1S_3\perp S_2S_4$ if and only if $AC=BD$.

Let $c_1,c_2,\ldots ,c_n,b_1,b_2,\ldots ,b_n$ $(n\geq 2)$ be positive real numbers. Prove that the equation \[ \sum_{i=1}^nc_i\sqrt{x_i-b_i}=\frac{1}{2}\sum_{i=1}^nx_i\] has a unique solution $(x_1,\ldots ,x_n)$ if and only if $\sum_{i=1}^nc_i^2=\sum_{i=1}^nb_i$.

Let $P,Q,R,S$ be the midpoints of the sides $BC,CD,DA,AB$ of a convex quadrilateral, respectively. Prove that \[4(AP^2+BQ^2+CR^2+DS^2)\le 5(AB^2+BC^2+CD^2+DA^2)\]