IrMO 1995

Points $A,X,D$ lie on a line in this order, point $B$ is on the plane such that $\angle ABX>120^{\circ}$ , and point $C$ is on the segment $BX$ . Prove the inequality:

$2AD \ge \sqrt{3} (AB+BC+CD)$ .

Let $S$ be the square consisting of all pints $(x,y)$ in the plane with $0\le x,y\le 1$ . For each real number $t$ with $0<t<1$ , let $C_t$ denote the set of all points $(x,y)\in S$ such that $(x,y)$ is on or above the line joining $(t,0)$ to $(0,1-t)$ . Prove that the points common to all $C_t$ are those points in $S$ that are on or above the curve $\sqrt{x}+\sqrt{y}=1$ .