In triangle CDE, point M lies on the side of CE, and the angle CMD is acute. Prove DE is more than DM.

For the proof, consider the triangle MDE, which turned out to be inside the triangle CDE. Since the angle DMC is acute, the adjacent angle DME is obtuse, that is, more than 90 degrees.
Now we consider the triangle MDE in terms of which sides are opposite to which angles.
The side DE lies opposite the obtuse angle DME, which we have already examined, the side MD is opposite the acute angle DEM (since there cannot be two obtuse angles in the triangle MDE.
Remember the rule: the larger side in the triangle is opposite the larger angle. This means that the side DE, located opposite the larger angle DМE, is larger than the side МD, which lies opposite the acute angle of MED.