Question 46
Let $n \ge 2$ be an integer. Determine the smallest positive real number $\gamma$ such that for any positive real numbers $x_1, x_2, \dots, x_n$ and any real numbers $y_1, y_2, \dots, y_n$ with $0 \le y_1, y_2, \dots, y_n \le \frac{1}{2}$ satisfying $x_1 + x_2 + \dots + x_n = y_1 + y_2 + \dots + y_n = 1$, we have
$$
x_1 x_2 \dots x_n \le \gamma (x_1 y_1 + x_2 y_2 + \dots + x_n y_n).
$$
$$
x_1 x_2 \dots x_n \le \gamma (x_1 y_1 + x_2 y_2 + \dots + x_n y_n).
$$