MathDB

3

Part of 2004 Czech and Slovak Olympiad III A

Problems(1)

121 chords on a circle

Source: Czech and Slovak third round,2004,p3

3/3/2012
Given a circle SS and its 121121 chords Pi(i=1,2,…,121)P_i (i=1,2,\ldots,121), each with a point Ai(i=1,2,…,121)A_i(i=1,2,\ldots,121) on it. Prove that there exists a point XX on the circumference of SS such that: there exist 2929 distinct indices 1≤k1≤k2≤…≤k29≤1211\le k_1\le k_2\le\ldots\le k_{29}\le 121, such that the angle formed by AkjX{A_{k_j}}X and Pkj{P_{k_j}} is smaller than 2121 degrees for every j=1,2,…,29j=1,2,\ldots,29.
combinatorics proposedcombinatorics