Problem. Let $n\in \mathbb{N}_{\ge 3}$.Let $0 \le a_1 \le a_2 \le \cdots \le a_n$. Prove or disprove that$$\sqrt{a_1a_2} + \sqrt{a_2a_3} + \ldots + \sqrt{a_na_1} \ge \sqrt[3]{a_1a_2a_3} + \sqrt[3]{a_2a_3a_4} + \ldots + \sqrt[3]{a_na_1a_2}.$$
I saw this problem recently which was originally asked 11 years ago (a Problem Statement Question (PSQ) - which does not meet the current quality standard ).
Some observations:
i) By letting $a_i = x_i^6, \forall i$, the desired inequality becomes$$\sum\limits_{\mathrm{cyc}} x_1^3 x_2^3\ge \sum\limits_{\mathrm{cyc}} x_1^2 x_2^2 x_3^2.$$
ii) @ivan investigated the case $n = 4, 5, 6$ and found that with the substitution$x_1 = y_1, x_2 = y_1 + y_2, \cdots, x_n = y_1 + y_2 + \cdots + y_n$,the polynomial $g(y_1, y_2, \cdots, y_n) := f(y_1, y_1 + y_2, \cdots, y_1 + y_2 + \cdots + y_n)$has only one negative coefficient(this term is $- y_1^4 y_2 y_n$).It suffices to prove that this is the case for $n\ge 7$.
iii) Based on @ivan's observation, let$$F(x_1, x_2, \cdots, x_n) := \sum\limits_{\mathrm{cyc}} x_1^3 x_2^3- \sum\limits_{\mathrm{cyc}} x_1^2 x_2^2 x_3^2- (x_2 - x_1 - x_n + x_{n-1})^2 x_1^4.$$Let$$G(y_1, y_2, \cdots, y_n):= F(y_1, y_1 + y_2, \cdots, y_1 + y_2 + \cdots + y_n).$$It suffices to prove that all coefficients of $G$ are non-negative.