5. What is the greatest number of acute angles that can occur in a convex polygon?
5. The maximum number of acute angles in a convex polygon is three. Indeed, the sum of all exterior angles of an $n$-sided polygon is always $4 d$. If any $n$-sided polygon ($n \geqslant 4$) had four acute interior angles, then the exterior angles supplementary to these interior angles would be obtuse, and their sum would be greater than $4 d$; therefore, the sum of all exterior angles would also be greater than $4 d$.