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Some Floors and Sawtooth Sum to Reals (Posted on 2023-09-14) Difficulty: 3 of 5
Determine the total number of real values of x satisfying this equation:

⌊x⌋ + 2 {-x} = 3x

• ⌊N⌋ is floor of N, which is equal to the greatest integer less than or equal to N.
{N} = N - ⌊N⌋.
• Computer program/excel solver assisted solutions are welcome, but a semi-analytic (hand calculator and p&p) methodology is preferred.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 1 of 2
-floor(-x)=ceil(x) which for nonintegers = floor(x)+1
and for integers = x

{-x} = -x - floor(-x) = -x+floor(x)+1 for nonintegers and for integers =0

so for integers the original equation can be written
which has integers solution 0 (and a noninteger -1/3 that we reject)

for nonintegers the original equation can be written
floor(x) + 2(-x+floor(x)+1)=3x
which has noninteger solutions -1/5 and 2/5 (and an integer solution 1 that we reject.)

So there are 3 solutions x={-1/5, 0, 2/5}

  Posted by Jer on 2023-09-14 10:39:39
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