Initially, the number $1$ and a non-integral number $x$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write $x^2$ on the blackboard in a finite number of moves?
Problem
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Tags: calculus, integration
ilikecake
03.09.2011 11:43
Umm... when the problem says, "In each step, we can choose two numbers on the blackboard, not necessarily different,..." does it mean that we can write a certain number more than once on the white board? For example, can I write 1+x twice on the whiteboard? Also when we find the difference of something, can it be in any order? For example, the difference of 1 and x can be $1-x$ or $x-1$.
Okay, so on the first step, we can find the sum and difference of 1 and x which is $x-1$* and $1+x$. Then we can find the reciprocal of each which looks like this: $\frac{1}{x-1}, \frac{1}{1+x}$. Then we can find the difference of then which is $\frac{x+1}{x^2-1}-\frac{x-1}{x^2-1}=\frac{2}{x^2-1}$. After this, we find the reciprocal of this which is $\frac{x^2-1}{2}$. *If possible, we can write the number twice, then find the sum which is the same as multiplying $\frac{x^2-1}{2}$ by 2 giving $x^2-1$. Finally we look back and add $x^2-1$ to 1 giving the wanted $x^2$.
*Step might not be possible based on the questions above