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Drifting Far Apart


Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n > m$. Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n - 2)$ and $(2m + 2)$ respectively. Find minimum integer $x$ with proof such that if $n - m \geq x$, then Pratyya's number will be larger than Payel's number everyday.


Proof Based Problems  


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Solution

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Suppose for some day, we have $n-m=k$. Then,

$n-m = k $

$2n-2m = 2k $

$2n-2-2m-2 = 2k-2-2 = 2k-4 $

$ (2n-2)-(2m+2) = 2k-4$

That means the difference will be $2k-4$ on the next day. From this, it's easy to see that when $k<4$, the difference will keep decreasing each day and eventually Payel's number will become larger. On the other hand, when $k\geq4$, the difference does not decrease. Therefore, our answer is $x=4$.

This is a proof based problem added for learning purposes and does not accept submissions.

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