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Points On A Circle II (Posted on 2010-07-13) Difficulty: 3 of 5
Refer to Points On A Circle.

(A) Seven points are placed on the circumference of a circle such that the distance between any two of the points, measured along the circumference, is an integer.

What is the smallest radius of the circle, given that each of the distances is unique?

(B) Seven points are placed on the circumference of a circle such that the distance between any two of the points, measured as a straight line, is an integer.

Determine the smallest radius of the circle. What is the smallest radius of the circle, given that it is rational?

Note: In Part (B) each of the distances may or may not be unique.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Part A - a further reduction | Comment 6 of 12 |
(In reply to Part A - a further reduction by Harry)

There is a problem with the set (1, 4, 2, 8, 9, 13, 11).
The arcs (13+11) and (1+4+2+8+9) each are 24 units in length.

If we increase your final distance from 11 to 20 -- (intermediate values having the same problem as your proposed set), we get a possible solution.
The 42 unique distances (from single arc lengths to the length of six consecutive arc lengths) are:
1, 2, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 27, 30, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 55, 56
This circumference of this circle is 57 units. This gives a radius of
57/(2pi) =~ 9.0718 units.

Edited on July 15, 2010, 4:51 am
  Posted by Dej Mar on 2010-07-15 04:32:47

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