1/1x2, 1/2x3, 1/ 3x4, ..., 1/2013 x 2014 on a circle
Source: 2014 Saudi Arabia Pre-TST 2.3
September 13, 2020
combinatorics
Problem Statement
The numbers
\frac{1}{1\times 2}, \frac{1}{2\times 3},\frac{1}{3\times 4},...,\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 .
(b) Prove that there exist ten consecutive numbers on the circle whose sum is less than .