MathLabs

Problem 2

For a positive integer kk, call an integer a pure kk-th power if it is mkm^k for some integer mm. Show that for every positive integer nn there exist nn distinct positive integers whose sum is a pure 20092009-th power and whose product is a pure 20102010-th power.
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ai=bisk2−1a_i=b_i s^{k^2-1}
Detailed analysis

Define ai=bisk2−1a_i=b_i s^{k^2-1}. They remain positive and distinct.