Track / Stage 6 / 206 of 400 #1206 of 2000

Problem 1206

National olympiad, first round

Algebra Difficulty 6.3 Prove it

2. Given that $a, b$ are positive numbers, and $n$ is a positive integer, prove: $\frac{a^{n}+b^{n}}{2} \geqslant\left(\frac{a+b}{2}\right)^{n}$ . (Former Soviet Union University Mathematics Competition Problem)

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

2. From the generalization of the Cauchy inequality, we get $\left.(F+1)(1+1) \cdots(F+1)\left(a^{n}+b^{n}\right) \geqslant(a+b)\right)^{n}$ , thus $\frac{a^{n}+b^{n}}{2} \geqslant\left(\frac{a+b}{2}\right)^{n}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.