[Beatmania]

$a^2+b^2+c^2 \geq \sum_{cyc}a\sqrt{b^2-bc+c^2}$

$\Leftrightarrow (a^2+b^2+c^2)^2 \geq (\sum_{cyc}a\sqrt{b^2-bc+c^2})^2$

$\Leftrightarrow \sum_{cyc}a^4+2\sum_{cyc}a^2b^2 \geq \sum_{cyc} [a^2b^2-a^2bc+a^2c^2]+2\sum_{cyc}ab\sqrt{(c^2-ca+a^2)(c^2-cb+b^2)}$

$\Leftrightarrow \sum_{cyc}a^4+\sum_{cyc}a^2bc \geq 2\sum_{cyc}ab\sqrt{(c^2-ca+a^2)(c^2-cb+b^2)}$

จาก Cauchy

$RHS \leq 2\sqrt{[\sum_{Cyc}abc^2-a^2bc+a^3b][\sum_{cyc}abc^2-ab^2c+ab^3]}$
$=2\sqrt{[\sum_{Cyc}a^3b][\sum_{cyc}ab^3]}\leq \sum_{Cyc}a^3b+\sum_{Cyc}ab^3\leq a^4+b^4+c^4+abc(a+b+c)=LHS$