Question 71
Find the largest constant $K\in\mathbb R$ with this property: if $a_1,a_2,a_3,a_4\gt 0$ satisfy, for all $i,j,k\in\mathbb N$ with $1\le i\lt j\lt k\le4$,
$$a_i^2+a_j^2+a_k^2\ge2(a_ia_j+a_ja_k+a_ka_i),$$
then
$$\begin{aligned}a_1^2+a_2^2+a_3^2+a_4^2\ge K(&a_1a_2+a_1a_3+a_1a_4\\&+a_2a_3+a_2a_4+a_3a_4).\end{aligned}$$
$$a_i^2+a_j^2+a_k^2\ge2(a_ia_j+a_ja_k+a_ka_i),$$
then
$$\begin{aligned}a_1^2+a_2^2+a_3^2+a_4^2\ge K(&a_1a_2+a_1a_3+a_1a_4\\&+a_2a_3+a_2a_4+a_3a_4).\end{aligned}$$