<s>[INST] $\sum_{cyc}^{}\frac{a(a+b+c)}{9b^2+(a+b+c)^2} \geq \frac{1}{2}$ I was attempting this question: https://www.youtube.com/watch?v=600X-ZGNBbk
Which is the following:
$$\sum_{cyc (a,b,c)}{}\frac{a}{b^2+1} \geq \frac{3}{2}$$
Where $a+b+c = 3$ and $a,b,c > 0 $
The video itself used a solution where they added $3$ and then subtracted $a+b+c$ to rewrite the inequality into something more manageable. Though I understand this solution, it seems arbitrary to me - thinking to add 3 and then subtract the condition is somewhat out of the blue.
What I did was to first try to homogenize the inequality to get rid of the condition. This meant writing $1$ as $\frac{(a+b+c)^2}{9}$ and multiplying the numerator by $\frac{a+b+c}{3}$:
$$\sum_{cyc}^{}\frac{\frac{a(a+b+c)}{3}}{b^2+\frac{(a+b+c)^2}{9}} \geq \frac{3}{2}$$
$$\leftrightarrow $$
$$\sum_{cyc}^{}\frac{a(a+b+c)}{9b^2+(a+b+c)^2} \geq \frac{1}{2}$$
I'm fairly certain this inequality is true without the condition of $a+b+c=3$, but I can't see a way to prove it. Multiplying out seems like a lot of trouble; is there an easier way (ideally with Olympiad mathematics methods)?
 [/INST] To prove your inequality (not the original inequality), I would introduce the condition $a + b + c = 1$. Note that
\begin{align*}
\frac{1}{9b^2+1} &\geq 1 - \frac{3}{2}b\\ 
\iff 0 &\leq \frac{3}{2}b\left(1 - 3b\right)^2
\end{align*}
(this is the Tangent Line Trick). Hence, we have
\begin{align*}
\sum_{\text{cyc}} \frac{a(a+b+c)}{9b^2+(a+b+c)^2} &= \sum_{\text{cyc}} \frac{a}{9b^2 + 1}\\
&\geq \sum_{\text{cyc}} a\left(1 - \frac{3}{2}b\right)\\
&= 1 - \frac{3}{2}(ab+bc+ca)\\
&\geq 1 - \frac{1}{2}(a+b+c)^2 && \text{by AM-GM}\\
&= \frac{1}{2}
\end{align*}
and so we're done.
</s>