2021 USAJMO Problems/Problem 1

Problem

Find all primes, with proof, that are the average of 2 consecutive primes.

Solution

After 2, all primes are odd, and the average of 2 and 3 is not an integer.

We proceed with mod 4. The 2 primes must be equivalent mod 4 because their difference cannot be 2 or 6 because that does not work.