The answer is (drum roll, please) July 16. Surprised? Not sure how to get that answer? Here's the solution.
When Albert looks at the list he says that Bernard won't be able to figure out the birthday. That rules out May and June as possibilities because both have unique numbers (18 and 19). If they were possibilities, Bernard could know the birthday just from the number Cheryl gave him.
Bernard, knowing that May and June are out, says he knows the birthday. That means the number can't be 14 because there are two of them, July 14 and August 14. So it has to be 15, 16 or 17.
Once Albert realizes that the number is 15, 16, or 17, he says he knows the birthday, too. If the month was August he wouldn't know because there are still two options, August 15 and August 17.
Therefore, the answer has to be July 16.
If you didn't get it, that's OK. This was problem 24 out of 25 for the contest and it was meant to stump hardcore math students. The Math Olympiad creators flat out said, "SASMO contests target the top 40% of the student population and the standards of most questions are just high enough to stretch the students."
Given how many people online are arguing the answer, or actively not caring, I would be surprised if Cheryl has ever had a birthday party.
So, did you get it right? Let me know in the comments.