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Find sum of distances on plane surface

Source: China TST 1998, problem 5

May 22, 2005
combinatoricsdistancecombinatorial geometrygeometrypoint set

Problem Statement

Let nn be a natural number greater than 2. ll is a line on a plane. There are nn distinct points P1P_1, P2P_2, …, PnP_n on ll. Let the product of distances between PiP_i and the other n−1n-1 points be did_i (i=1,2,i = 1, 2, …, nn). There exists a point QQ, which does not lie on ll, on the plane. Let the distance from QQ to PiP_i be CiC_i (i=1,2,i = 1, 2, …, nn). Find Sn=∑i=1n(−1)n−ici2diS_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}.
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