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Set of Lines with a Condition (Posted on 2010-08-11) Difficulty: 2 of 5
S is a set of lines such that every line in S either passes through point P or has slope M.
Find all the possible values of number of lines that S can have if total number of points of intersection of all lines in S is 49.

No Solution Yet Submitted by Praneeth    
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solution Comment 1 of 1
first, let x be the number of elements of S which intersect P, let y be the number of elements of S which have slope M but do not intersect P. let z be the number of elements of S which intersect P and have slope M.

Now my solution is based on the assumption that the elements of S are unique, or in other no 2 elements of S coincide.  If the assumption holds then the number of intersections are given by (x-z)*y if x=1 or 0 and (x-z)*y+1 if x>1

this is because all of the y lines are parallel and thus have no intersections with each other.  Each of the x lines going through P (without slope M) intersect each of the y parallel lines once.  Now obviously z is either 0 or 1
if z=0 we have
x=1 y=49 giving us 50 lines
or x>1
xy+1=49
xy=48
giving us (x,y): (2,24), (3,16), (4,12), (6,8), (8,6), (12,4), (16,3), (24,2), (48,1)
giving us line totals of 26, 19, 16, 14 and 49
 
if z=1 we have
(x-1)*y+1=49
(x-1)*y=48
(x,y): (2,48), (3,24), (4,16), (5,12), (7,8), (9,6), (13,4), (17,3), (25,2), (49,1)
giving line totals of
27, 20, 17, 15, 14, 50

putting this all together the possible number of lines are
14, 15, 16, 17, 19, 20, 26, 27, 49, 50
for 10 different values

  Posted by Daniel on 2010-08-11 20:08:26
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