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June 16, 2020

Non-Parallel lines | PRMO 2017 | Question 22

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


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

Non-Parallel lines

for 10 lines are 20

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

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