Learn Russian: Студенты часто начинают недосыпать к началу сессии. - Students often start getting too little sleep by the beginning of finals.

The prefix недо- indicates that an action is performed insufficiently or falls below a normal, healthy requirement. Here, it attaches to the root for sleep to create недосыпать (to get too little sleep, to be sleep-deprived). This is the exact opposite of the prefix пере- for excessive action, which gives us пересыпать (to oversleep).

Wait, doesn't сыпать usually mean to pour? How did it end up in a word about sleep?

It looks identical, but it is a completely different root! When you add a prefix to спать (to sleep), forming the imperfective aspect requires adding the suffix -ы-, turning the root into -сыпать. So, спать becomes the perfective недоспать, which then becomes the imperfective недосыпать.

Why is недосыпать strictly in the imperfective aspect in this sentence?

It must be imperfective because it immediately follows the verb начинают (start). In Russian grammar, phase verbs that show the beginning, continuation, or end of an action (like начинать, продолжать, заканчивать) always force the following infinitive to be in the imperfective aspect.

Why is к началу used here to mean by the beginning instead of до начала?