Try this beautiful problem from the Pre-RMO, 2017 based on Non-Parallel Lines.
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?
Polygon
Non-parallel lines
Integers
But try the problem first...
Answer: is 36.
PRMO, 2017, Question 22
Elementary Number Theory by David Burton
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.
Try this beautiful problem from the Pre-RMO, 2017 based on Non-Parallel Lines.
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?
Polygon
Non-parallel lines
Integers
But try the problem first...
Answer: is 36.
PRMO, 2017, Question 22
Elementary Number Theory by David Burton
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.
