AMATYC Student Mathematics League

I have some test questions from last semester of the AMATYC Student Mathematics League.
it totally has 20 questions. I did some but still feel confused by some questions.
one is :
The equation a^6+b^2+c^2=2009 has a solution in positive integers a, b, and c in which exactly two of a,b, and c are powers of 2. Find a+b+c.
The choicesfor this one is:
A.43. B.45. C.47 D. 49 E.51.
And another one is :
For all integers k>=0,P(k)=(2^2+2^1+1)(2^2-2^1+1)(2^4-2^2+1)...( 2^2^k+1-2^2^k+1)-1 is always the product of two integers n and n+1. Find the smallest value of k for which n+(n+1)>=10^1000
The choices for this one is:
A.9 B. 10 C.11 D.12 E.13

I really worked on these two hard, but I still feel very awkward to them...