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-

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 + \dots + T_{n}

S = 2 + 4 + 7 + 11 + \dots + T_{n}

0 = 2 + [2 + 3 + 4 + \dots (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 .