Suppose in the plane 10 pairwise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?


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Question Number 21097 by Tinkutara last updated on 12/Sep/17

$Suppose in the plane 10 pairwise$ $nonparallel lines intersect one another .$ $What is the maximum possible number$ $of polygons (with finite areas) that can$ $be formed ?$
	Answered by Tinkutara last updated on 15/Sep/17

	$L e t u s a s s u m e a_{n} r e p r e s e n t s t h e$ $n u m b e r o f r e g i o n s o f p l a n e f o r m e d$ $b y p a i r w i s e n o n p a r a l l e l l i n e s .$ $T h e n T_{1} = 2, T_{2} = 4, T_{3} = 7, T_{4} = 11 a n d s o o n .$ $L e t S = 2 + 4 + 7 + 11 + . . . + T_{n}$ $S = 2 + 4 + 7 + 11 + . . . + T_{n}$ $0 = 2 + [2 + 3 + 4 + . . . (n - 1) t e r m s] - T_{n}$ $T_{n} = 1 + \frac{n (n + 1)}{2}$ $∴ T_{10} = 1 + \frac{10 \times 11}{2} = 56$ $S e e i n g t h e n u m b e r o f n o n o v e r l a p p i n g$ $p o l y g o n s f o r m e d b y u p t o 4 l i n e s,$ $w e g e t n u m b e r o f p o l y g o n s, N = T_{n} - 2 n .$ $∴ N u m b e r o f p o l y g o n s f o r m e d b y$ $10 n o n p a r a l l e l l i n e s = 56 - 20 = 36 .$
	Commented by Tinkutara last updated on 15/Sep/17

	$Question is exactly the same as it$ $appeared in PRMO 2017 . I think$ $somewhere is miswording in question .$ $But since all answers are 2 - digit integers$ $in this test, so 36 can be a possibility .$
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