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Getting Floored With 28 (Posted on 2008-09-06) Difficulty: 2 of 5
The greatest integer ≤ Y is denoted by [Y] , and {Y} = Y - [Y].

How many distinct real Y satisfy this equation, whenever 1 ≤ Y ≤ 28 ?

                                         {Y2} = {Y}2

Note: While a solution may be trivial with the aid of a computer program, show how to derive it without one.

See The Solution Submitted by K Sengupta    
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Solution re: solution | Comment 3 of 4 |
(In reply to solution by Charlie)

Reading Paul's comment made me go back and look at mine. I now recognize that the crossover that falls "somewhere between sqrt(3) and 2", for example, is actually at the endpoint that's only approached.

The actual numbers in the intervals, that are added together, are 2, 4, 6, 8, ..., 54, or 

Sigma [y=1 to 27] (2y)

and that this is 27^2 + 27.

Then one more must be added for the rightmost endpoint, making 27^2+28, agreeing with Paul's answer.

  Posted by Charlie on 2008-09-07 02:18:55
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