TIFR CSE 2025 | Part A | Question: 14

Question

Let $c_{1}, c_{2}, \ldots, c_{n}$ and $z$ be complex numbers such that
\[
\frac{1}{z-c_{1}}+\frac{1}{z-c_{2}}+\ldots+\frac{1}{z-c_{n}}=0
\]
Assume that the numbers $c_{1}, c_{2}, \ldots, c_{n}$ are represented in the complex plane by the vertices of a convex $n$-gon $C$. Then

• A. the number $z$ always lies strictly outside $C$.
• B. the number $z$ always lies inside or on $C$.
• C. the number $z$ may lie inside or outside $C$ depending on $c_{1}, c_{2}, \ldots, c_{n}$.
• D. if $z$ lies inside $C$ then $z$ must be the centroid, i.e., $z=\frac{1}{n} \sum_{k=1}^{n} c_{k}$.
• E. None of the above