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Inscribed circle and Line Choice (Posted on 2010-06-27) Difficulty: 3 of 5
(A) Out of 100 straight lines having respective lengths 1, 2, 3, ......, 99, 100; determine the total number of ways in which four straight lines may be chosen which will form a quadrilateral in which a circle may be inscribed.

(B) Out of 101 straight lines having respective lengths 1, 2, 3, ......, 99, 100, 101; determine the total number of ways in which four straight lines may be chosen which will form a quadrilateral in which a circle may be inscribed.

No Solution Yet Submitted by K Sengupta    
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Solution No Subject Comment 3 of 3 |
Starting with n=4, I get 1,3,7,13,22,34,50,... as the number of quadrilaterals.  This sequence can be derived from taking sums of series alternate triangular numbers:
1 3 6 10 15 21 28...
    1  3  6 10 15...
          1  3  6...
                1...
--------------------
1 3 7 13 22 34 50...
There is also a general formula for the nth term: t(n+2) = (4*n^3+6*n^2-4*n+3*(-1)^n-3)/48

Then to solve for 100 and 101 lines, substitute n=98 and 99 to get t(100)=79625 and t(101) = 82075.

OEIS has this sequence as A002623.

  Posted by Brian Smith on 2016-02-02 11:17:12
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