Russian problems Problem 99 The difference between the longest and shortest diagonals of the regular n-gon equals its side. Find all possible n. Solution
: n = 9. For n  7 the side has length 2R sin pi/n, the shortest diagonal has length 2R sin 2pi/n, and the longest diagonal has length 2R for n even and 2R cos pi/2n for n odd (where R is the radius of the circumcircle). Thus we require: sin 2pi/n + sin pi/n = 1 and n even, or sin 2pi/n + sin pi/n = cos pi/2n and n odd. Evidently the lhs is a strictly decreasing function of n and the rhs is an increasing function of n, so there can be at most one solution of each equation. The second equation is satisfied by n = 9, although it is easier to see that there is a quadrilateral with the longest diagonal and shortest diagonals as one pair of opposite sides, and 9-gon sides as the other pair of opposite sides. The angle between the longest side and an adjacent side is 60, so that its length is the length of the shortest diagonal plus 2 x 9-gon side x cos 60. Hence that is the only solution for n odd. For n = 8 we have the same quadrilateral as for the 9-gon except that the angle is 67.5 and hence the difference is less than 1. For n = 10, sin 2pi/10 + sin pi/10 = sin pi/10 (2 cos pi/10 + 1) = 10, and hence no solutions for n even. Russian 91-100 (C) John Scholes jscholes@kalva.demon.co.uk 1 Oct 1998