Question 151
Let $a, b, c, d$ be nonnegative real numbers not exceeding $1$. Prove that
$$
\frac{1}{1+a+b} + \frac{1}{1+b+c} + \frac{1}{1+c+d} + \frac{1}{1+d+a} \le \frac{4}{1+2\sqrt[4]{abcd}}.
$$
$$
\frac{1}{1+a+b} + \frac{1}{1+b+c} + \frac{1}{1+c+d} + \frac{1}{1+d+a} \le \frac{4}{1+2\sqrt[4]{abcd}}.
$$