Protons' repulsion within a nucleus
How positively charged protons remain glued to each other while they should repel each other out of nucleus?
Due to one of the four fundamental interactions known as the strong force which, at the scale involved is roughly two orders of magnitude stronger than electromagnetic interaction which repels the protons.
It is interesting to note that small nuclei have roughly equal number of protons and neutrons while heavy nuclei have excess of neutrons over protons. The reason for this change in the ratio of protons to neutrons is that, unlike electromagnetic interaction, the strong force has very limited range and in heavy nuclei greater separation of protons increases the relative significance of the destabilizing electromagnetic force.