CZ-SK-MO 2004 Problem 3

Given a circle $S$ and its $121$ chords $P_i (i=1,2,\ldots,121)$ , each with a point $A_i(i=1,2,\ldots,121)$ on it. Prove that there exists a point $X$ on the circumference of $S$ such that: there exist $29$ distinct indices $1\le k_1\le k_2\le\ldots\le k_{29}\le 121$ , such that the angle formed by ${A_{k_j}}X$ and ${P_{k_j}}$ is smaller than $21$ degrees for every $j=1,2,\ldots,29$ .