Question 20
The 2013 numbers
$$
\frac{1}{1 \times 2}, \frac{1}{2 \times 3}, \frac{1}{3 \times 4}, \ldots, \frac{1}{2013 \times 2014}
$$
are arranged randomly on a circle.
a. Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{4000}$.
b. Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{10000}$.
$$
\frac{1}{1 \times 2}, \frac{1}{2 \times 3}, \frac{1}{3 \times 4}, \ldots, \frac{1}{2013 \times 2014}
$$
are arranged randomly on a circle.
a. Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{4000}$.
b. Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{10000}$.