A certain set of positive integers
(a,b,c) causes the sum of their reversals (i.e. S=1/a+1/b+1/c) to be the closest to 1/2 without equaling or exceeding it.
Find S.
Explain your method.
(In reply to
re: solution (to tedious to do manually) by Steve Herman)
Allowing for equality allowed only a few extra values and the ones larger than 1/3+1/7+1/43 were all equal to 1/2:
1/3+1/12+1/12
2
3
3 3
3 4
3 5
3 6
3 7
3 7 42 0.5
3 7 43 0.499446290143965
3 8
3 8 24 0.5
3 8 25 0.498333333333333
3 9
3 9 18 0.5
3 9 19 0.497076023391813
3 10
3 10 15 0.5
3 10 16 0.495833333333333
3 11
3 11 14 0.495670995670996
3 12
3 12 12 0.5
3 12 13 0.493589743589744
4
4 4
4 5
4 5 20 0.5
4 5 21 0.497619047619048
4 6
4 6 12 0.5
4 6 13 0.493589743589744
4 7
4 7 10 0.492857142857143
4 8
4 8 8 0.5
4 8 9 0.486111111111111
5
5 5
5 5 10 0.5
5 5 11 0.490909090909091
5 6
5 6 8 0.491666666666667
6
6 6
6 6 6 0.5
6 6 7 0.476190476190476
2
3
3 3
3 4
3 5
3 6
3 7
3 7 42 0.5
3 7 43 0.499446290143965
3 8
3 8 24 0.5
3 8 25 0.498333333333333
3 9
3 9 18 0.5
3 9 19 0.497076023391813
3 10
3 10 15 0.5
3 10 16 0.495833333333333
3 11
3 11 14 0.495670995670996
3 12
3 12 12 0.5
3 12 13 0.493589743589744
4
4 4
4 5
4 5 20 0.5
4 5 21 0.497619047619048
4 6
4 6 12 0.5
4 6 13 0.493589743589744
4 7
4 7 10 0.492857142857143
4 8
4 8 8 0.5
4 8 9 0.486111111111111
5
5 5
5 5 10 0.5
5 5 11 0.490909090909091
5 6
5 6 8 0.491666666666667
6
6 6
6 6 6 0.5
6 6 7 0.476190476190476
Edited on May 15, 2020, 11:02 am
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Posted by Charlie
on 2020-05-15 10:59:52 |
re(2): solution (to tedious to do manually)
