, 2015

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Let us call a point $(m, n)$ a lattice point if both $m$ and $n$ are integers. Call a lattice point $(m, n)$ a good point if it lies in the portion of the $xy$-plane given by $1 \le x \le 20$, $1 \le y \le 15$.
If a straight line $\ell$ lying on the $xy$-plane is parallel either to the $x$-axis or to the $y$-axis, then there are only $15$ (or $20$) straight lines parallel to $\ell$ and going through a good point. Therefore, the slope of the line $\ell$ satisfying the condition of the problem must be a real number not equal to $0$. Next, suppose the slope of the line $\ell$ is an irrational number. Then, any straight line parallel to $\ell$ can go through at most one good point, since a line connecting any pair of lattice points must either be parallel to the $y$-axis or have a rational slope. Therefore, there are $300$ straight lines parallel to $\ell$, going through one good point, which contradicts the assumption of the problem. Thus, we can assume that the line $\ell$ satisfying the assumption of the problem has the slope of the form $\pm \frac{b}{a}$, where $a$, $b$ are integers not equal to $0$ and are relatively prime.
If there are lattice points on the line with slope $\pm \frac{b}{a}$, then they are located on the line with gaps $(a, b)$, since $a$ and $b$ are relatively prime. So, if either $|a| > 20$ or $|b| > 15$, then all the lines parallel to $\ell$ going through different good points are distinct. Since there are $300$ good points, this contradicts the assumption. Therefore, we must have both $|a| \le 20$ and $|b| \le 15$. Let us first consider the case where $a, b > 0$. In this case no pair of lattice points from the set $\{(m', n') \mid \text{either } 1 \le m' \le a \text{ or } 1 \le n' \le b \}$ can lie on the same straight line parallel to line $\ell$, since on such lines lattice points are located with gaps $(a, b)$. On the other hand for any lattice point $(m'', n'')$ from the set $\{(m'', n'') \mid a < m'' \le 20, \text{ and } b < n'' \le 15\}$ the straight line through this point and parallel to line $\ell$ must go through a lattice point $(m', n')$ satisfying either $1 \le m' \le a$ or $1 \le n' \le b$. (See the diagram below.) Consequently, we see that there are exactly $20 \times 15 - (20 - |a|)(15 - |b|)$ straight lines parallel to $\ell$ going through a good point. When $a$ or $b$ or both are negative, we can argue in the same way to conclude that the number of lines satisfying the requirement of the problem is $20 \times 15 - (20 - |a|)(15 - |b|)$.

($k$ is a positive integer)

By assumption, we have $20 \times 15 - (20 - |a|)(15 - |b|) = 222$, from which we obtain
$(20 - |a|)(15 - |b|) = 78$. There is only one way, i.e., $78 = 6 \times 13$, to express the
number $78$ as the product of a positive integer less than equal to $20$ and a positive
integer less than or equal to $15$. If we set $20 - |a| = 6$ and $15 - |b| = 13$, then we
get $|a| = 14$ and $|b| = 2$, which will contradict the assumption that $a$ and $b$ are
relatively prime. So, we must have $20 - |a| = 13$ and $15 - |b| = 6$, which yield
$|a| = 7$ and $|b| = 9$.
The slope of a line perpendicular to line $l$ is $\frac{-a}{b}$. Note that both of the conditions $|b| \le 20$ and $|-a| \le 15$ are satisfied. Then we can check that exactly same arguments applied for lines parallel to $l$ as above can be applied to the lines perpendicular to $l$. Thus we can conclude that the number we seek for the problem is
$$20 \times 15 - (20 - |b|)(15 - |-a|) = 222.$$