NO-ABEL 1999

An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$ . Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$ , and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$ . The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$ , respectively. (a) Prove that triangle $AFG$ is equilateral. (b) Find the ratio between the areas of triangles $AFE$ and $ABC$ .

For every nonempty subset $R$ of $S = \{1,2,...,10\}$ , we define the alternating sum $A(R)$ as follows: If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$ . (a) Is it possible to partition $S$ into two sets having the same alternating sum? (b) Determine the sum $\sum_{R} A(R)$ , where $R$ runs over all nonempty subsets of $S$ .