Question 23
Prove that there exist integers $x_1, x_2, \dots, x_{10}, y_1, y_2, \dots, y_{10}$ satisfying:
(1) For $i = 1, 2, \dots, 10$, $|x_i| \le 10^{10}$ and $|y_i| \le 10^{10}$;
(2) The point set in the plane
$$
X = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) \middle| a_1, a_2, \dots, a_{10} \in \{0, 1\} \right\}
$$
contains exactly 1024 distinct points;
(3) For any two parallel lines in the plane at distance 1, the strip region between them (including the lines) contains at most two points from $X$.
(1) For $i = 1, 2, \dots, 10$, $|x_i| \le 10^{10}$ and $|y_i| \le 10^{10}$;
(2) The point set in the plane
$$
X = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) \middle| a_1, a_2, \dots, a_{10} \in \{0, 1\} \right\}
$$
contains exactly 1024 distinct points;
(3) For any two parallel lines in the plane at distance 1, the strip region between them (including the lines) contains at most two points from $X$.