All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

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

⌊x⌋ + 2 {-x} = 3x

Notes:
• ⌊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. Comment 2 of 2 |
Analytic solution then computer verification.

Let x = i + f
where i is ⌊x⌋  and f is {x}

Case 1:  x = 0; 0 is a solution

Case 2:  x > 0
⌊x⌋ + 2{-x} = 3x becomes
i + 2*(1-f) = 3i + 3f
2 = 2i + 5f --> i=0, f=2/5
x = 0.4

Case 3:  x < 0
x=-(i+f), ⌊x⌋=-(i+1), {-x}=f
-(i+1) + 2f = -3(i+f)
-1 = -2i - 5f
if i=0, f=0.2
x = -(i+f) = -0.2

Solutions:  -0.2, 0, 0.4

------------
import math
epsilon = .00000001

for i in range(-100000,100000):
n = i/1000
if abs(math.floor(n) + 2*((-n)%1) - 3*n) < epsilon:
print(n)

Output:
-0.2
0.0
0.4

 Posted by Larry on 2023-09-14 11:27:25

 Search: Search body:
Forums (0)
Random Problem
Site Statistics
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox: