AMC8 2011 8, Problem 21

Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?

Solution: If at least half the guesses are too low, then Norb's age must be greater than $36.$ If two of the guesses are off by one, then his age is in between two guesses whose difference is $2.$ It could be $31,37,$ or $48,$ but because his age is greater than $36$ it can only be $37$ or $48.$ Lastly, Norb's age is a prime number so the answer must be $\fbox{37}$

Answer: 37