, 2024

Let $O$ be the origin, $A$ be the point with coordinates $(20,24)$, and $B$ be the point with coordinates $(4, 202)$.

The slope of $OA$ is $\frac{24}{20} = 1.2$. The slope of $OB$ is $\frac{202}{4} = 50.5$.

[[IMAGE0]]

The line with equation $y = mx$
passes through $(0,0)$.

If $m < 0$, the line with equation
$y = mx$ does not pass through the
first quadrant.

If $0 \leq m < 1.2$, the line with
equation $y = mx$ is less steep than
$OA$ and so does not intersect line
segment $AB$.

If $m > 50.5$, the line with
equation $y = mx$ is steeper than
$OB$ and so does not intersect line
segment $AB$.

For integers $m$ with $2 \leq m \leq 50$, the line with equation
$y = mx$ will intersect the line
segment $AB$.

There are $49$ such integers $m$.