Question 81
Let $x_1, x_2, ..., x_{14}$ be non-negative numbers, sum of which is equal to $1$. Prove that:
$$
x_1x_2x_3x_4 + x_2x_3x_4x_5 + \dots + x_{11}x_{12}x_{13}x_{14} + x_{12}x_{13}x_{14}x_1 + x_{13}x_{14}x_1x_2 + x_{14}x_1x_2x_3 \le \frac{1}{4^4}. \quad \text{(Serhiy Zhyliaiev)}
$$
$$
x_1x_2x_3x_4 + x_2x_3x_4x_5 + \dots + x_{11}x_{12}x_{13}x_{14} + x_{12}x_{13}x_{14}x_1 + x_{13}x_{14}x_1x_2 + x_{14}x_1x_2x_3 \le \frac{1}{4^4}. \quad \text{(Serhiy Zhyliaiev)}
$$