Question 6
Let $s_1, s_2, s_3, \ldots$ be an infinite,
nonconstant sequence of rational numbers,
meaning it is not the case that $s_1 = s_2 = s_3 = \ldots$.
Suppose that $t_1, t_2, t_3, \ldots$ is also an infinite,
nonconstant sequence of rational numbers
with the property that $(s_i - s_j)(t_i - t_j)$
is an integer for all $i$ and $j$.
Prove that there exists a rational number $r$
such that $(s_i - s_j)r$ and $(t_i - t_j)/r$
are integers for all $i$ and $j$.
Proposed by Gabriel Carroll (credit in the supplied third-party notes).
nonconstant sequence of rational numbers,
meaning it is not the case that $s_1 = s_2 = s_3 = \ldots$.
Suppose that $t_1, t_2, t_3, \ldots$ is also an infinite,
nonconstant sequence of rational numbers
with the property that $(s_i - s_j)(t_i - t_j)$
is an integer for all $i$ and $j$.
Prove that there exists a rational number $r$
such that $(s_i - s_j)r$ and $(t_i - t_j)/r$
are integers for all $i$ and $j$.
Proposed by Gabriel Carroll (credit in the supplied third-party notes).