<s>[INST] Given positives $a, b, c$ such that $a + b + c = 3$, prove that $\sum_{cyc}\frac{1}{a^2 + 4b^2 + c^2} \le \frac{1}{2}$. 
Given positives $a, b, c$ such that $a + b + c = 3$, prove that $$\frac{1}{c^2 + 4a^2 + b^2} + \frac{1}{a^2 + 4b^2 + c^2} + \frac{1}{b^2 + 4c^2 + a^2} \le \frac{1}{2}$$

We have that $$a^2 + 4b^2 + c^2 = a^2 + (a + b + c + 1)b^2 + c^2 = (b^2 + a)a + (b + 1)b^2 + (b^2 + c)c$$
$$\implies \sum_{cyc}\frac{1}{a^2 + 4b^2 + c^2} = \frac{1}{(a + b + c)^2} \cdot \sum_{cyc}\frac{(a + b + c)^2}{(b^2 + a)a + (b + 1)b^2 + (b^2 + c)c}$$
$$ \le \frac{1}{(a + b + c)^2} \cdot \sum_{cyc}\left(\frac{a}{b^2 + a} + \frac{1}{b + 1} + \frac{c}{b^2 + c}\right)$$
Furthermore, $\dfrac{a}{c^2 + a} + \dfrac{c}{a^2 + c} = 1 - \dfrac{(c + a - 2)ca}{(c^2 + a)(a^2 + c)}$, $$\sum_{cyc}\frac{1}{a^2 + 4b^2 + c^2} \le \frac{1}{(a + b + c)^2} \cdot \sum_{cyc}\left[\frac{1}{b + 1} - \frac{(c + a - 2)ca}{(c^2 + a)(a^2 + c)}\right] + \frac{1}{3}$$
Then I don't know what to do next.
 [/INST] Another way.
We'll prove that our inequality is true for any real $a$, $b$ and $c$ such that $a+b+c=3.$
Indeed, let $a+b+c=3u$, $ab+ac+bc=3v^2,$ where $v^2$ can be negative, and $abc=w^3$.
Thus, by my first proof we need to prove that:
$$\sum_{sym}(a^6-4a^5b+13a^4b^2-2a^4bc-6a^3b^3-12a^3b^2c+10a^2b^2c^2)\geq0$$ or
$$27w^6+A(u,v^2)w^3+B(u,v^2)\geq0,$$ where $A$ and $B$ are polynomials of $u$ and $v^2$.
We'll prove that even the following inequality is true.
$$\sum_{sym}(a^6-4a^5b+13a^4b^2-2a^4bc-6a^3b^3-12a^3b^2c+10a^2b^2c^2)\geq$$
$$\geq\frac{1}{3}\left(\sum_{cyc}(a^3-a^2b-a^2c+abc)\right)^2.$$
Since $$\sum_{cyc}(a^3-a^2b-a^2c+abc)=27u^3-27uv^2+3w^3-9uv^2+3w^3+3w^3$$ and $$27-\frac{1}{3}\cdot9^2=0,$$ we see that the last inequality is a linear inequality of $w^3$.
Thus, it's enough to prove the last inequality for extreme value of $w^3$, 
which happens for equality case of two variables.
Since the last inequality is homogeneous, it's enough to assume $b=c=1$, which gives
$$2(a-1)^4(a^2+5)\geq\frac{1}{3}(a-1)^4(a+2)^2$$ or
$$(a-1)^4(5a^2-4a+26)\geq0,$$ which is obvious.
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