Question 18
Let $(u_{jk})$, $1 \le j \le 3$, $1 \le k \le 2$ be real numbers and let $N$ be an integer such that
$$
\max_{1 \le k \le 2} \sum_{j=1}^{3} |u_{jk}| \le N.
$$
Suppose $M$ and $l$ are positive integers such that $l^2 \lt (M+1)^3$. Prove that there exist integers $\xi_1, \xi_2, \xi_3$, not all zero, such that
$$
\max_{1 \le j \le 3} |\xi_j| \le M \quad \text{and} \quad \left| \sum_{j=1}^{3} u_{jk} \xi_j \right| \le \frac{MN}{l}, \quad \text{for } k = 1, 2.
$$
$$
\max_{1 \le k \le 2} \sum_{j=1}^{3} |u_{jk}| \le N.
$$
Suppose $M$ and $l$ are positive integers such that $l^2 \lt (M+1)^3$. Prove that there exist integers $\xi_1, \xi_2, \xi_3$, not all zero, such that
$$
\max_{1 \le j \le 3} |\xi_j| \le M \quad \text{and} \quad \left| \sum_{j=1}^{3} u_{jk} \xi_j \right| \le \frac{MN}{l}, \quad \text{for } k = 1, 2.
$$