Problem 2.
For a positive integer k, call an integer a pure k-th power if it can
be represented as m^k for some integer m. Show that for every positive
integer n there exist n distinct positive integers such that their sum
is a pure 2009-th power, and their product is a pure 2010-th power.
Set a1,...,an distinct.
N = a1^2010 + a2^2010 + ... + an^2010
Define Xi = N^(2010*2008) ai^2010
Then
X1+X2+...+Xn = N^(2010*2008) * N = (N^2009)^2009, a pure 2009-th power
X1X2...Xn = (N^2008n a1a2...an) ^ 2010, a pure 2010-th power...
好像太简单了...