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Small Steps

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

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Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

The symbol [ ] means 'the integer part of '.

Consider the three numbers

$$[2x];\ 2[x];\ [x + {1\over 2}] + [x - {1\over 2}]$$

Can they ever be equal?

Can they ever take three different values?