VWO 1997

Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there's no better solution.

In the cartesian plane, consider the curves $x^2+y^2=r^2$ and $(xy)^2=1$ . Call $F_r$ the convex polygon with vertices the points of intersection of these 2 curves. (if they exist)

(a) Find the area of the polygon as a function of $r$ . (b) For which values of $r$ do we have a regular polygon?

$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$ . Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$ , Call the summed area of the first $k$ triangles $A_k$ . Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$ , going through the points $b_2$ and $a_2$ as shown on the picture. (yes, $cd$ is parallel to $a_1b_1$ there)

Show $A_k < S$ for every positive integer $k$ .