H. Asia Pacific 2010 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.

import Mathlib.Algebra.BigOperators.Group.Finset.Defs
import Mathlib.Data.Fintype.Basic

theorem solution (n : ℕ) (hn : 0 < n) : ∃ a : Fin n → ℕ, (∀ i, 0 < a i) ∧
    a.Injective ∧ (∃ b, ∑ i, a i = b ^ 2009) ∧ (∃ c, ∏ i, a i = c ^ 2010) := sorry

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