Y by langkhach11112, rmtf1111, Adventure10, Mango247
Let
be positive real numbers,
. Denote by
their geometric mean, and by
the sequence of arithmetic means defined by
Let
be the geometric mean of
. Prove the inequality
and establish the cases of equality.
Proposed by Finbarr Holland, Ireland




![\[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \]](http://latex.artofproblemsolving.com/6/f/4/6f49684d75b0c4d0055613f1cf6bcafb034de25f.png)


![\[
n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \]](http://latex.artofproblemsolving.com/8/6/4/8644f80b9729df06edaf0de781fc98ae16392151.png)
Proposed by Finbarr Holland, Ireland