Question 66
a. Let $a_1, a_2, \dots, a_n \gt 0$. Prove that $a_1^{a_1} a_2^{a_2} \cdots a_n^{a_n} \ge (a_1 a_2 \cdots a_n)^{\frac{a_1+a_2+\cdots+a_n}{n}}$.
b. Let $x, y, z \ge 0$ and $x + y + z = 1$. Prove that $0 \le xy + yz + zx - 3xyz \le \frac{1}{4}$.
b. Let $x, y, z \ge 0$ and $x + y + z = 1$. Prove that $0 \le xy + yz + zx - 3xyz \le \frac{1}{4}$.