<s>[INST] If $a\leq (b + c)/2$ with $a,b,c>0$, why $a^2\leq (b^2 + c^2)/2$? If $a\leq (b + c)/2$ with $a,b,c>0$, why $a^2\leq \frac{b^2 + c^2}{2}$? I can only see how to get $a^2\leq \frac{b^2+c^2 + 2ab}{4}$.
 [/INST] That's not true. Let $a = 9, b = 5, c =5$. Then $9 \leq 5 +5$ but $81 \not\leq 25 + 25$.
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