O is the center of a regular n-gon P1P2 ... Pn and X is a point outside the n-gon on the line OP1. Show that XP1 XP2 ... XPn + OP1n = OXn.
Evidently it is sufficient to prove the result for the case OP1 = 1. So represent Pk by the complex number ωk, where ω = ei 2π/n an nth root of unity. Represent X by the real number r. Then we have to show that |r - 1||r - ω| ... |r - ωn-1| = rn - 1, but that follows immediately because rn - 1 = (r - 1)(r - ω) ... (r - ωn-1).
15th Putnam 1955
© John Scholes
26 Nov 1999