6

Part of 2013 Serbia National Math Olympiad

Problems(1)

The largest constant K with given propery and 4 real numbers

Source: Serbian National Olympiad 2013, Problem 6

Find the largest constant

K\in \mathbb{R}

with the following property: if

a_1,a_2,a_3,a_4>0

are numbers satisfying

a_i^2 + a_j^2 + a_k^2 \geq 2 (a_ia_j + a_ja_k + a_ka_i)

1\leq i<j<k\leq 4

a_1^2+a_2^2+a_3^2+a_4^2 \geq K (a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).

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