11th Eötvös Competition Problems 1904

1.  A pentagon inscribed in a circle has equal angles. Show that it has equal sides. 2.  Let a be an integer, and let p(x1, x2, ... , xn) = ∑1n k xk. Show that the number of integral solutions (x1, x2, ... , xn) to p(x1, x2, ... , xn) = a, with all xi > 0 equals the number of integral solutions (x1, x2, ... , xn) to p(x1, x2, ... , xn) = a -