This is trickier than it seems.

By statement 1, A needs to know that B doesn't know. How? Well, B can only know for sure if B sees an 18 or 19.

• If A sees May, B could have 15, 16, or 19. If it's 19, B would definitely know. Thus it can't be May.
• If A sees June, B could have 17 or 18, so it can't be June.
• If A sees July, B could have 14 or 16, so it could be July.
• If A sees August, B could have 14, 15, or 17 so it could be August.

By statement 2, B must now know. B knows it could only be July or August.

• If B had 14, he wouldn't be able to know still.
• If B has 16, he knows it's July 16. If B has 15 or 17, he knows it's August 15 or August 17.

By statement 3, A must now know.

• If B had 15 or 17 and A had August, A still wouldn't know the exact date in August.
• Therefore, it must be July 16.

EDIT: after all, this was not for primary school kids. It was for a math olympiad, in Singapore, for 14/15 year olds...

WHEW