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Rounding Machine

Problem

A particular number machine works as follows.


E.g. 3.4 maps 6.8 maps 6 or 7.9 maps 15.8 maps 15

A different number machine does the following.


E.g. 3.4 maps 3 maps 6 or 7.9 maps 7 maps 14

Notice that 3.4 came out as 6 from both machines, whereas 7.9 came out differently. What must be special about a number for the same value to come out of each machine?


Solution

If x is the value going into each machine, the machines can be expressed as [2x] and 2[x] respectively.

All numbers of the form n.m under the integer part function will become n by definition.

Therefore 2[n.m] = 2n (i.e. independent of m)

But, [2 times n.m] = [2 times (n + m/10)] = [2n + m/5]

If m less than 5, 0 less than or equal m less than 1 implies [2n + m/5] = 2n, whereas for m greater than or equal 5, 1 less than or equal m less than2 implies [2n + m/5] = 2n + 1.

And so the decimal part of x must be less than .5 for 2[x] to be equal to [2x].

When is [x + 0.5] + [x minus 0.5] equal to 2[x] and [2x]?

Problem ID: 97 (Jan 2003)     Difficulty: 2 Star

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