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, 2015

Combinatorics Difficulty 7.5 National Olympiad, round 2 Prove it Baltic Way

The numbers
$\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, \dots, \frac{2015}{2016}$
are written on a blackboard. With each move, one may erase any two numbers $a$ and $b$ and replace them with
$3ab - 2a - 2b + 2.$
What will be the single remaining number after 2014 moves?

Solution

Note that if $a = \frac{1344}{2016} = \frac{2}{3}$ , then
$3ab - 2a - 2b + 2 = \frac{2}{3},$
irrespective of the value of $b$ . Hence, $\frac{2}{3}$ will always remain on the blackboard. $\square$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.