IMO 1979 SL 20

Origin: SWE

Problem

Given the integer n > 1 and the real number a > 0 determine the maximum of n−1 i=1 xixi+1 taken over all nonnegative numbers xi with sum a.

Solution

Let xk = max{x1, x2, . . . , xn}. Then xixi+1 \leqxixk for i = 1, 2, . . . , k −1 and xixi+1 \leqxkxi+1 for i = k, . . . , n −1. Summing up these inequalities for i = 1, 2, . . ., n −1 we obtain n−1  i=1 \leqxk(x1 + \cdot \cdot \cdot + xk−1 + xk+1 + \cdot \cdot \cdot + xn) = xk(a −xk) \leqa2 4 . We note that the value a2/4 is attained for x1 = x2 = a/2 and x3 = \cdot \cdot \cdot = xn = 0. Hence a2/4 is the required maximum.