<s>[INST] $a+b+c+d=1, a,b,c,d ≠ 0$, then prove that $(a + \frac{1}{a})^2 + (b + \frac{1}{b})^2 + (c + \frac{1}{c})^2 + (d + \frac{1}{d})^2 \ge \frac{289}{4} $ If $a+b+c+d=1$, $a,b,c,d ≠ 0$, prove that $$\left(a + \frac{1}{a}\right)^2 + \left(b + \frac{1}{b}\right)^2 + \left(c + \frac{1}{c}\right)^2 + \left(d + \frac{1}{d}\right)^2 \ge \frac{289}{4} $$
I tried expanding the entire LHS and I got 
$a^2 + b^2 + c^2 + d^2 + \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} + \frac{1}{d^2} + 8$ on the LHS. Then I tried applying AM-GM to the first 4 terms and the last 4 terms and I got 
$4 \sqrt{abcd} + \frac{4}{\sqrt{abcd}}$ but that didnt seem useful.
Any hints?
Thank you in advance.
 [/INST] Let $$f(x)=x^2+\frac{1}{x^2}.$$
Thus, $f$ is a convex function on $(0,+\infty)$ and $f$ is a convex function on $(-\infty,0)$.
We need to consider the following cases.


*

*$a\geq b\geq c\geq d>0$. 


The case  $0>a\geq b\geq c\geq d$ is impossible because $a+b+c+d=1.$
Now, our inequality follows from the Jensen's inequality:
$$8+\sum_{\text{cyc}}\left(a^2+\frac{1}{a^2}\right)\geq8+4\left(\left(\frac{a+b+c+d}{4}\right)^2+\frac{1}{\left(\frac{a+b+c+d}{4}\right)^2}\right)=\frac{289}{4}.$$


*$a\geq b\geq c>0>d$ (the case $a>0>b\geq c\geq d$ is a similar).

*$a\geq b>0>c\geq d$.
In the last case we obtain a counterexample: $$(a,b,c,d)=\left(-1,-1,\frac{3}{2},\frac{3}{2}\right).$$ 
In this case $$\sum_{\text{cyc}}\left(a+\frac{1}{a}\right)^2=\frac{313}{18}<\frac{289}{4},$$ which says that your inequality is wrong! 
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