1. [4] Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n, n k)$ in the $(x, y)$ plane. Suppose that two markers whose $x$ coordinates differ by 4 have distance 31. What is the distance between the markers at $(7,7 k)$ and $(19,19 k)$ ?
Answer: 93 The difference of the $x$-coordinates of the markers is $12=3 \cdot 4$. Thus, by similar triangles (where we draw right triangles whose legs are parallel to the axes and whose hypotenuses lie along the line $y=k x$ ), the distance between the markers is $3 \cdot 31=93$.