Problems

Prove that $S_{ABCD}\le (AB\times BC+AD\times DC)/2$.

Prove that $\mathrm{\angle }ABC>{90}^{\circ }$ if and only if the point $B$ lies inside a circle with diameter $AC$.

The radii of two circles are $R$ and $r$, and the distance between their centres is equal $d$. Prove that these circles intersect if and only if $|R-r|<d<R+r$.

On a line segment of length 1, $n$ points are given. Prove that the sum of the distances from some point out of the ones on the segment to these points is no less than $n/2$.