Question 36
Let $a$, $b$, $c$, $d$ be positive real numbers with $a + b + c + d = 4$. Prove the inequality
$$
\frac{(a + \sqrt{b})^2}{\sqrt{a^2 - ab + b^2}} + \frac{(b + \sqrt{c})^2}{\sqrt{b^2 - bc + c^2}} + \frac{(c + \sqrt{d})^2}{\sqrt{c^2 - cd + d^2}} + \frac{(d + \sqrt{a})^2}{\sqrt{d^2 - da + a^2}} \le 16.
$$
$$
\frac{(a + \sqrt{b})^2}{\sqrt{a^2 - ab + b^2}} + \frac{(b + \sqrt{c})^2}{\sqrt{b^2 - bc + c^2}} + \frac{(c + \sqrt{d})^2}{\sqrt{c^2 - cd + d^2}} + \frac{(d + \sqrt{a})^2}{\sqrt{d^2 - da + a^2}} \le 16.
$$