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Ten people are sitting in a row, and each is thinking of a negative integer no smaller than $-15$. Each person subtracts, from his own number, the number of the person sitting to his right (the rightmost person does nothing). Because he has nothing else to do, the rightmost person observes that all the differences were positive. Let $x$ be the greatest integer owned by one of the 10 people at the beginning. What is the minimum possible value of $x$?

Not sure how to go about this. I think it is 1 since -14-(-15)=1. I'm not sure though.

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  • $\begingroup$ Yes, they are integers no smaller than -15 $\endgroup$ – ddswsd Feb 21 '18 at 18:25
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Let $x_1,x_2,\dots,x_{10}$ be your numbers. Then $y_i=x_i-x_{i+1}.$ for $i=1,\dots,9,$ be the results of the subtractions. Since $y_i$ are positive, then $x_i\geq 1+x_{i+1}.$ In particular, $x_1$ is the largest value, and $x_1\geq x_{10}+9\geq -15+9=-6.$

Now you just need to find such an example where $x_1=-6.$ That is you minimum largest $x_i.$

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As I understand it, we shall have
$-15 \le x_n \le -1$ and $0 < x_n -x_{n+1}$.

That is
$-15 \le x_{10}<x_9< \cdots < x_1 \le -1$

Then greatest $x$, means $x_n \le -1$ for any $n$, while "minimum possible" means that when $x_{10}=-15$ then $x_1=-6$ (?)

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  • $\begingroup$ I'm not sure I understand this. $\endgroup$ – ddswsd Feb 21 '18 at 18:25
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Each person's number has to be greater than the number of the person sitting to his right, so that when he subtracts his righthand person's number from his own, he gets a positive number. This means that the person on the far left had the largest number to start with; in other words, the leftmost person had the number $x$. This also means that the rightmost person had the smallest integer to start with.

We want to minimize $x$, so it makes sense to give the rightmost person $-15$, since that's the lowest number he could have started with. We also want to make the differences as small as possible, so let all the differences be 1 (remember, all the numbers must be integers). This means that as we go from right to left, we add 1 for each person we pass. Since there were 9 differences calculated in total (one person did nothing), we add 9 overall in going from the rightmost person to the leftmost person. Since the rightmost person had $-15$, the minimum that the leftmost person could have had is $-15+9=\boxed{-6}$.

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