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I'm going to a math competition soon, and there are a few problems that have stumped both me and my teacher. I was wondering if any of you could help me out?

Prove

(a^2+b^2)/(a+b) >= sqrt(ab)

for all real A and B.

p(x)mod(x-19) = 99

p(x)mod(x-99) = 19

p(x)mod(x-19)(x-99) = ?

Suppose for an integer, n, there exists numbers 2n+1 and 3n+1 that are perfect squares. Prove that n cannot be prime.

Thanks a ton for the help! I appreciate it.