If non negative real numbers $a_1$, $a_2$$\dots$, $a_n$ satisfy $\sum\limits_{i=1}^na_i=1$, prove that\[\sum_{1\le i<j\le n}a_ia_j\left(a_i^2+a_j^2\right)\le\frac18.\]
I think we can use induction because $\left(\frac12,\frac12,0,\dots,0\right)$ and its permutations are (possibly not the only?) equality cases. So we start with $3$-var.
We need to prove $8\sum\limits_{\rm cyc}\left(a_1^3a_2+a_1a_2^3\right)\le\left(\sum\limits_{\rm cyc}a_1\right)^3$, or \[8\sum\limits_{\rm cyc}a_1^2a_2a_3+\left(\sum\limits_{\rm cyc}a_1^2-2\sum\limits_{\rm cyc}a_1a_2\right)^2\ge0~,\] (by the undetermined coeffitients), which is true.
For the induction step, assume $n-1$-var version is proven, we need to consider the $n$-var version. I tried to apply the inductive assumption on the $n$ numbers \[a_1,a_2\dots,a_{n-2},a_{n-1}+a_n,\]but it failed.