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UkraineMO

Algebra Difficulty 6.0 AIME, harder Prove it Ukraine

Petryk programmed calculator in such a way that if there is number xx on the screen and then the button «=>» is pressed, number x−1x+1\frac{x-1}{x+1} appears on the screen. Petryk pressed the button «=>» 2014 times, and then 2016 appeared on the screen. What number was on the screen at the beginning? The screen can show not only integer numbers.

Solution

Let us look how the number changes after the button «=>» is pressed. Let the first number be xx. Then after the first pressing we have x−1x+1\frac{x-1}{x+1}.

After the second we have
x−1x+1−1x−1x+1+1=x−1−x−1x−1+x+1=−1x. \frac{\frac{x-1}{x+1}-1}{\frac{x-1}{x+1}+1} = \frac{x-1-x-1}{x-1+x+1} = -\frac{1}{x}.
After the third we have
−1x−1−1x+1=−1−x−1+x. \frac{-\frac{1}{x}-1}{-\frac{1}{x}+1} = \frac{-1-x}{-1+x}.
After the fourth we have
−1−x−1+x−1−1−x−1+x+1=−1−x+1−x−1−x−1+x=x. \frac{-\frac{1-x}{-1+x}-1}{\frac{-1-x}{-1+x}+1} = \frac{-1-x+1-x}{-1-x-1+x} = x.
Thus, after every fourth pressing the number that was at the beginning appears on the screen. Therefore, after 2012 pressings we have xx. After 2 more pressings we obtain −1x=2016-\frac{1}{x} = 2016, thus x=−12016x = -\frac{1}{2016}.

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