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If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.

i p=2 then $2|p+2q$. So p cannot be 2.
Let p be the largest awesome prime.
The least value of q for any p is 1. So, we can write-

$ p+2\equiv1,2 (mod 3) $
now suppose,
$ p+2\equiv1 (mod 3) $
so, $p+4\equiv0 (mod 3) $
so, $2q=4$
and so, $q=2$. which is not possible.

So, for the largest awesome prime-
$ p+2\equiv2 (mod 3) $
or, $ p+6\equiv0 (mod 3) $
or, $ p\equiv0 (mod 3)$
Now we can say, p is a prime and divisible by 3. So the only value of $p=3$

And it's the largest awesome prime.

If $p>3$ then $p,p+2,p+4$ forms a complete set of residues mod $3$. Thus one of then is divisible by $3$. So, $3$ is the largest awesome prime.