MathDB

4

Part of 2011 Korea National Olympiad

Problems(2)

Korea Second Round 2011

Source: Korea Second Round 2011 #4

8/21/2011
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.
combinatorics unsolvedcombinatorics
Korea Second Round 2011

Source: Korea Second Round 2011 #8

8/21/2011
Let x1,x2,⋯ ,x25 x_1, x_2, \cdots, x_{25} real numbers such that 0≤xi≤i(i=1,2,⋯ ,25) 0 \le x_i \le i (i=1, 2, \cdots, 25) . Find the maximum value of x13+x23+⋯+x253−(x1x2x3+x2x3x4+⋯x25x1x2)x_{1}^{3}+x_{2}^{3}+\cdots +x_{25}^{3} - ( x_1x_2x_3 + x_2x_3x_4 + \cdots x_{25}x_1x_2 )
functioninequalities proposedinequalities