Try this ... from AIMO 2007 Intermediate Paper
Find a prime, p, with the property that for some larger prime number, q, both 2q - p and 2q + p are prime numbers. Prove that there is only one such prime p.
I don't even know where to start!
Any prime p > 3 is of the form $\displaystyle 6k\pm1$ for some integer $\displaystyle k\ge1$. For p > 3, you can try all combinations of $\displaystyle p=6k_1\pm1$ and $\displaystyle q=6k_2\pm1$ and verify that $\displaystyle 2q\pm p$ is never prime.
If p = 2, then 2q+p and 2q-p are both even and neither is equal to p, so neither is prime.
Hence p = 3 is the only prime satisfying the given property.