IMO 1967 LL ROM47

Origin: ROM

Problem

Prove the inequality x1x2 \cdot \cdot \cdot xk
xn−1

• xn−1
• \cdot \cdot \cdot + xn−1 k  \leqxn+k−1 +xn+k−1 +\cdot \cdot \cdot+xn+k−1 k , where xi > 0 (i = 1, 2, . . ., k), k \inN, n \inN.

Solution

Using the A–G mean inequality we get (n + k −1)xn 1x2 \cdot \cdot \cdot xk \leqnxn+k−1

• xn+k−1
• \cdot \cdot \cdot + xn+k−1 k , (n + k −1)x1xn 2 \cdot \cdot \cdot xk \leqxn+k−1
• nxn+k−1
• \cdot \cdot \cdot + xn+k−1 k , . . . . . . . . . . . . . . . (n + k −1)x1x2 \cdot \cdot \cdot xn k \leqxn+k−1
• xn+k−1
• \cdot \cdot \cdot + nxn+k−1 k . By adding these inequalities and dividing by n + k −1 we obtain the desired one. Remark. This is also an immediate consequence of Muirhead’s inequality.