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Distinct Values Determination (Posted on 2015-10-20) Difficulty: 3 of 5
It is known that X is a real number with 0 ≤ X ≤ 100

Find the total count of distinct values that can be assumed by this expression:

floor(X) + floor(2*X) + floor(5*X /3) + floor(3*X) + floor(4*X)

No Solution Yet Submitted by K Sengupta    
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Solution Is the answer an airplane model? (Spoiler?) | Comment 1 of 4
Min value of 0 occurs when x = 0
Max value of 1166 occurs when x = 100
So, the number of distinct possible values is less than or equal to 1167.

Are all of those values achievable?  No, they are not.

For instance, when x = 1, the expression equals 1*10 + floor(5/3) = 11.
But, when x is slightly less than 1 it is 4 less. So, 10 and 9 and 8 are not achievable.
This occurs whenever x is an integer not divisible by 3, so it is the case for x = 1,2,4,5,7,8...97,98, 100.
All together, (67 values of x) * 3 = 201 impossible values

The situation is slightly different when x is divisible by 3.
For instance, when x = 3, the expression equals 3*10 + floor(5*3/3) = 35.
But, when x is slightly less than 3 it is 5 less. So, 34 and 33 and 32 and 31 are not achievable.
All together, (33 values of x) * 4 = 132 impossible values.

Also, 2 and 4 are not relatively prime, so we lose some values when x is a multiple of 1/2 but not 1.
For instance, when x = 1/2, expression = 0 + 1 + 0 + 1 + 2 = 4
But, when x is slightly less than 1/2 it is 2 less. So, 3 is not achievable.
All together, (100 values of x) * 1 = 100 impossible values.

Total achievable values = 1167 - 201 - 132 - 100 = 734

Edited on October 20, 2015, 7:27 pm
  Posted by Steve Herman on 2015-10-20 08:32:46

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