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Korea Second Round 2011

Source: Korea Second Round 2011 #4

August 21, 2011
combinatorics unsolvedcombinatorics

Problem Statement

Let k,nk,n be positive integers. There are knkn points P1,P2,⋯ ,PknP_1, P_2, \cdots, P_{kn} on a circle. We can color each points with one of color c1,c2,⋯ ,ck c_1, c_2, \cdots , c_k . In how many ways we can color the points satisfying the following conditions?
(a) Each color is used n n times.
(b) ∀i≠j \forall i \not = j , if Pa P_a and Pb P_b is colored with color ci c_i , and Pc P_c and Pd P_d is colored with color cj c_j , then the segment PaPb P_a P_b and segment PcPd P_c P_d doesn't meet together.
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