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Who is Mr Boole?

Source: 2017 China South East Mathematical Olympiad Day1 P2

July 30, 2017
functionmodular arithmeticalgebranumber theory

Problem Statement

Let xi∈{0,1}(i=1,2,⋯ ,n)x_i \in \{0,1\}(i=1,2,\cdots ,n),if the value of function f=f(x1,x2,⋯ ,xn)f=f(x_1,x_2, \cdots ,x_n) can only be 00 or 11,then we call ff a nn-var Boole function,and we denote Dn(f)={(x1,x2,⋯ ,xn)∣f(x1,x2,⋯ ,xn)=0}.D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}. (1)(1) Find the number of nn-var Boole function; (2)(2) Let gg be a nn-var Boole function such that g(x1,x2,⋯ ,xn)≡1+x1+x1x2+x1x2x3+⋯+x1x2⋯xn(mod2)g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2, Find the number of elements of the set Dn(g)D_n(g),and find the maximum of n∈N+n \in \mathbb{N}_+ such that ∑(x1,x2,⋯ ,xn)∈Dn(g)(x1+x2+⋯+xn)≤2017.\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.
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