Question 113
Let $a$, $b$, $c$, $d$ be four real numbers such that $a \geqslant b \geqslant c \geqslant d \gt 0$ and $a + b + c + d = 1$. Prove that
$$
(a + 2b + 3c + 4d) a^{a} b^{b} c^{c} d^{d} \lt 1
$$
$$
(a + 2b + 3c + 4d) a^{a} b^{b} c^{c} d^{d} \lt 1
$$