isi-entrance 2005 Q10

isi-entrance · India · solved Number Theory Combinatorial Number Theory and Counting
a) $n$ lines are drawn through a point $A$ inside a circle, creating chords. Show that the number of regions created inside the circle is $(n+1)^2$, assuming no three lines meet at any point other than $A$.
b) Using the result from part a), find the total number of regions when additional lines are drawn, and show the total is $(n+1)^2 + (2n+1)n$.
a) For $n=1$: 1 line through $A$ creates 2 regions. For general $n$: $n$ lines through $A$ create $(n+1)$ regions on each side, giving $(n+1)^2$ regions total.
b) Drawing line $AB'$: triangle $ABC$ equivalent configuration gives $(n+1)$ regions. Drawing line $AD$ generates $(2n+1)$ more regions (as there are $2n+2$ lines). Adding $(n-2)$ more lines gives $n$ additional groups. Total regions $= (n+1)^2 + (2n+1)n$. 
a) $n$ lines are drawn through a point $A$ inside a circle, creating chords. Show that the number of regions created inside the circle is $(n+1)^2$, assuming no three lines meet at any point other than $A$.

b) Using the result from part a), find the total number of regions when additional lines are drawn, and show the total is $(n+1)^2 + (2n+1)n$.
Paper Questions
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