MathLabs

Problem 5

A set of 2n+1 points in the plane has no three collinear and no four concyclic. A circle divides the set if it passes through 3 of the points and has exactly n-1 points inside it. Prove that the number of circles dividing the set is even if and only if n is even.
Step 2 of 5: Track how the interior count changes
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Xi∈R⟺Xi lies inside Cj (i<j),Xj∈L⟺Xj lies inside CiX_i\in R\Longleftrightarrow X_i\text{ lies inside }C_j\ (i<j),\quad X_j\in L\Longleftrightarrow X_j\text{ lies inside }C_i
Detailed analysis

For i<j, the circle Ci contains the points of Cj on the R side, while Cj contains the points of Ci on the L side. Therefore, if fi is the number of points inside Ci, then fi+1=fi when Xi and Xi+1 lie on opposite sides; fi+1=fi-1 when both lie in L; and f(i+1)=f(i)+1 when both lie in R.