9b555 u need to confirm
determine all pairs (h,s) of positive integers with the following property
if one draws s horizontal lines and another s lines which satisfy
1)they r not horizontal
2)no two of them are parallel
3)no three of h+s are concurrent,
then the number of regios formed by these h+s lines is 1992
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4 Answers
9A similar q was asked by MAK (see an interesting question 1)
the q makes sense if it reads as
if one draws h horizontal lines and another s lines which satisfy .....
now h parralel lines intoduce h+1 regions
after that its a trivial observation that whenever a line intersects it introduces x +1 new regions where x is no of intersections .
so tot no of regions are
== h+1 (due to h parallel lines) + h+1 + h+2 + .........h+s ( as there are s non parralel lines and ive considered a pt of intersection only once)
== (s+1)(h+ s/2) +1
ie 1991 = (s+1)(h + s/2)= 11 X 181 both are prime
hence s is 10 and h is 176
(176,10 ) is only pair possible !!!
