Try this beautiful problem from the Pre-RMO, 2017 based on Non-Parallel Lines.

## Non Parallel Lines – PRMO 2017

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

- is 107
- is 36
- is 840
- cannot be determined from the given information

**Key Concepts**

Polygon

Non-parallel lines

Integers

## Check the Answer

But try the problem first…

Answer: is 36.

PRMO, 2017, Question 22

Elementary Number Theory by David Burton

## Try with Hints

First hint

Let f(x) be defined as f(x+1)=f(x)+(x+1), f(1)=2 where x is number of lines drawn in a plane which makes the plane into f(x) number of regions

f(2)=f(1)+2=4

f(3)=f(2)+3=7

f(4)=f(3)+4=11

Second Hint

f(5)=f(4)+5=16

f(6)=f(5)+6=22

f(7)=f(6)+7=29

f(8)=f(7)+8=37

f(9)=f(8)+9=46

f(10)=f(9)+10=56

Final Step

open region for three lines are 6

for 10 lines are 20

so, number of non-overlapping polygon=56-20=36.

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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