<s>[INST] $1/(51 +u^2) + 1/(51 +v^2) + 1/(51 +w^2)$ inequality I need to prove that $$1/(51 +u^2) + 1/(51 +v^2) + 1/(51 +w^2) \leq 1/20$$ given $u + v + w = 9$ and $u,v,w$ positive reals.
Using AM-HM inequality with $(51 +u^2, 51 +v^2, 51 +w^2)$ I arrive at 
$(51 +u^2 + 51 +v^2 + 51 +w^2)/3 \geq 3/((1/(51 +u^2) +1/(51 +v^2) 1/(51 +w^2))$, $$or,$$ 
$1/(51 +u^2) +1/(51 +v^2)+1/(51 +w^2)  \leq 9/(51 +u^2 + 51 +v^2 + 51 +w^2)$
In order to take the next step, I need to estimate the RHS with an upper bound. Here is where I face the problem. With $(u+v+w)^2 \leq 3(u^2 + v^2 + w^2)$ from  C-S inequality and $u+v+w =9$ we get  $27 \leq  (u^2 + v^2 + w^2)$ which does not provide an upper bound. I tried several other inequalities, but each lead to a bound on the "wrong" side. Am I at all looking in the right direction? 
 [/INST] You wrote the following:
"Using AM-HM inequality with $(51 +u^2, 51 +v^2, 51 +w^2)$ I arrive at 
$(51 +u^2 + 51 +v^2 + 51 +w^2)/3 \geq 3/((1/(51 +u^2) +1/(51 +v^2) 1/(51 +w^2))$, $$or,$$ 
$1/(51 +u^2) +1/(51 +v^2)+1/(51 +w^2)  \leq 9/(51 +u^2 + 51 +v^2 + 51 +w^2)$"
See please better, the last line should be:
$1/(51 +u^2) +1/(51 +v^2)+1/(51 +w^2)  \geq 9/(51 +u^2 + 51 +v^2 + 51 +w^2)$
You proved that
$$\sum_{cyc}\frac{1}{51+u^2}\geq\frac{9}{\sum\limits_{cyc}(51+u^2)},$$ which does not help for the proof of your inequality.
By the way, the Tangent Line method helps.
Indeed, we need to prove that
$$\frac{1}{20}-\sum_{cyc}\frac{1}{51+u^2}\geq0$$ or
$$\sum_{cyc}\left(\frac{1}{60}-\frac{1}{51+u^2}\right)\geq0$$ or
$$\sum_{cyc}\frac{(u-3)(u+3)}{51+u^2}\geq0$$ or
$$\sum_{cyc}\left(\frac{(u-3)(u+3)}{51+u^2}-\frac{u-3}{10}\right)\geq0$$ or
$$\sum_{cyc}\frac{(u-3)^2(7-u)}{51+u^2}\geq0,$$ which is true for $\max\{u,v,w\}\leq7.$
Let $u>7$.
Thus, $$\sum_{cyc}\frac{1}{51+u^2}<\frac{1}{51+7^2}+\frac{2}{51}<\frac{1}{20}$$
and we are done!
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