BG-MO 2000

In the coordinate plane, a set of $2000$ points $\{(x_1, y_1), (x_2, y_2), . . . , (x_{2000}, y_{2000})\}$ is called good if $0\leq x_i \leq 83$ , $0\leq y_i \leq 83$ for $i = 1, 2, \dots, 2000$ and $x_i \not= x_j$ when $i\not=j$ . Find the largest positive integer $n$ such that, for any good set, the interior and boundary of some unit square contains exactly $n$ of the points in the set on its interior or its boundary.

Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$ . Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$ , where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$ , and for each $a, b \in A$ , the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$ , $c \in A$ satisfy $a+ b + c = 0$ , then $f(a)+ f(b) + f(c) = 0$ .