Question 188
For positive real numbers $a$, $b$, $c$ and $d$ such that $a^2 + b^2 + c^2 + d^2 = 1$ prove that
$$
a^2 b^2 c d + a b^2 c^2 d + a b c^2 d^2 + a^2 b c^2 d + a^2 b c d^2 + a b^2 c d^2 \le \frac{3}{32},
$$
and determine the cases of equality.
$$
a^2 b^2 c d + a b^2 c^2 d + a b c^2 d^2 + a^2 b c^2 d + a^2 b c d^2 + a b^2 c d^2 \le \frac{3}{32},
$$
and determine the cases of equality.