A particular number machine works as follows.
E.g. 3.4 6.8 6 or 7.9 15.8 15
A different number machine does the following.
E.g. 3.4 3 6 or 7.9 7 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?
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 n.m] = [2 (n + m/10)] = [2n + m/5]
If m 5, 0 m 1 [2n + m/5] = 2n, whereas for m 5, 1 m 2 [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 0.5] equal to 2[x] and [2x]?