Question 50
Let $a$, $b$, $c$, $d$ be non-negative real numbers such that
$$
\frac{1}{a+3} + \frac{1}{b+3} + \frac{1}{c+3} + \frac{1}{d+3} = 1.
$$
Prove that there is a permutation $(x_1, x_2, x_3, x_4)$ of the sequence $(a, b, c, d)$ such that
$$
x_1x_2 + x_2x_3 + x_3x_4 + x_4x_1 \ge 4.
$$
$$
\frac{1}{a+3} + \frac{1}{b+3} + \frac{1}{c+3} + \frac{1}{d+3} = 1.
$$
Prove that there is a permutation $(x_1, x_2, x_3, x_4)$ of the sequence $(a, b, c, d)$ such that
$$
x_1x_2 + x_2x_3 + x_3x_4 + x_4x_1 \ge 4.
$$