The two parts? A statement and a prophecy, but more than that; the statement can also be read as a goal, an instruction, a way of life and a description - a path to follow if you will. The prophecy is not 'do this and you shall get this', it is more subtle than that, it is saying by doing this (being aware of our spiritual poverty, practising meekness etc) we share in the Lord (the blessing) and the second part comes naturally.
The initial statements are describing folks' actions, choices they make. Choosing to be aware of our own spiritual poverty; choosing to mourn for our own and the world's hurts, choosing to hunger and thirst for righteousness - these are not natural 'states', Jesus is saying is saying two (at least) things:
1. Choosing to do these things is the right way, the way of the Law, the Law that Jesus is bringing a new and correct interpretation of, and by doing them you share in the blessing of the Lord. So he is saying 'Act', and it is no coincidence that the Sermon on the Mount is bookended by calls to action: Matthew 7: 24-27 - Everyone then who hears these words and acts on them will be like a wise man who built his house on rock....
2. Feeling these things is good. If you do feel the pain of the world, if you feel merciful (and surely many of his audience did at many a point in their lives) then you are right and follow the Law and I am here to tell you so. This last point is amplified later in his description of hypocrites,
Looked at this way (and it is not the only valid way to read the Beatitudes) , there is no paradox there is only God's blessing.