IMO 1979 SL 11

Origin: GDR

Problem

Given real numbers x1, x2, . . . , xn (n \geq2), with xi \geq1/n (i = 1, 2, . . ., n) and with x2 1 +x2 2 +\cdot \cdot \cdot+x2 n = 1, find whether the product P = x1x2x3 \cdot \cdot \cdot xn has a greatest and/or least value and if so, give these values.

Solution

Let us define yi = x2 i . We thus have y1 + y2 + \cdot \cdot \cdot+ yn = 1, yi \geq1/n2, and P = \sqrty1y2 . . . yn. The upper bound is obtained immediately from the AM–GM inequality: P \leq1/nn/2, where equality holds when xi = \sqrtyi = 1/\sqrtn. For the lower bound, let us assume w.l.o.g. that y1 \geqy2 \geq\cdot \cdot \cdot \geqyn. We note that if a \geqb \geq1/n2 and s = a + b > 2/n2 is fixed, then ab = (s2 −(a −b)2)/4 is minimized when |a −b| is maximized, i.e., when b = 1/n2. Hence y1y2 \cdot \cdot \cdot yn is minimal when y2 = y3 = \cdot \cdot \cdot = yn = 1/n2. Then y1 = (n2 −n + 1)/n2 and therefore Pmin = \sqrt n2 −n + 1/nn.