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Turkish NMO 2012 - P5

Source: Turkish NMO 2nd Round 2012 - P5

November 29, 2012
inductioninequalitiescombinatorics proposedcombinatorics

Problem Statement

Let PP be the set of all 20122012 tuples (x1,x2,…,x2012)(x_1, x_2, \dots, x_{2012}), where xi∈{1,2,…20}x_i \in \{1,2,\dots 20\} for each 1≤i≤20121\leq i \leq 2012. The set A⊂PA \subset P is said to be decreasing if for each (x1,x2,…,x2012)∈A(x_1,x_2,\dots ,x_{2012} ) \in A any (y1,y2,…,y2012)(y_1,y_2,\dots, y_{2012}) satisfying yi≤xi(1≤i≤2012)y_i \leq x_i (1\leq i \leq 2012) also belongs to AA. The set B⊂PB \subset P is said to be increasing if for each (x1,x2,…,x2012)∈B(x_1,x_2,\dots ,x_{2012} ) \in B any (y1,y2,…,y2012)(y_1,y_2,\dots, y_{2012}) satisfying yi≥xi(1≤i≤2012)y_i \geq x_i (1\leq i \leq 2012) also belongs to BB. Find the maximum possible value of f(A,B)=∣A∩B∣∣A∣⋅∣B∣f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}, where AA and BB are nonempty decreasing and increasing sets (∣⋅∣\mid \cdot \mid denotes the number of elements of the set).
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