To divide a line segment PQ in the ratio m:n, where m and n are positive integers, what is the minimum number of points that should be marked on the ray PX, given that ∠PQX is an acute angle?

The minimum number of points that should be marked on the ray PX is m + n - 1.

To divide the line segment PQ in the ratio m : n, we draw a ray PX from P such that ∠PQX is an acute angle. Then, we mark points on the ray PX at equal distances. 

Let the distance between each marked point be x. We can then find the length of PX by using the fact that the distance ratio between PQ and PX is equal to the ratio m : n.

Therefore, we have:

$\frac{\mathrm{PQ}}{\mathrm{PX}} = \frac{\mathrm{m}}{\mathrm{n}}$

$\frac{\mathrm{PQ}}{(\mathrm{PQ}+\mathrm{x}(\mathrm{m}+\mathrm{n}\left)\right)} = \frac{\mathrm{m}}{\mathrm{n}}$

$\mathrm{nPQ} = \mathrm{m}(\mathrm{PQ} + \mathrm{x}(\mathrm{m}+\mathrm{n}\left)\right)$

$\mathrm{nPQ} = \mathrm{mPQ} + \mathrm{mx} + \mathrm{mnx}$

$\left(\mathrm{PQ}\right)(\mathrm{n}-\mathrm{m}) = \mathrm{x}\left(\mathrm{mn}\right)$

$\mathrm{x} = \frac{\left(\mathrm{PQ}\right)(\mathrm{n}-\mathrm{m})}{\mathrm{mn}}$

Since we want x to be a positive integer, we choose the smallest such value, which is $\mathrm{x} = \frac{\left(\mathrm{PQ}\right)(\mathrm{n}-\mathrm{m})}{\mathrm{mn}}$, rounded up to the nearest integer. The minimum number of points is then the number of intervals between the points plus 1, which is m + n - 1.