Let \(\{a_i\}_{i=1}^m\) and \(\{b_i\}_{i=1}^n\) be non-negative integer sequences such that all elements in \(\{a_i\}_{i=1}^m \cup \{b_i\}_{i=1}^n\) are pairwise distinct.

Prove there are no solutions to \[\displaystyle\sum_{i=1}^m a_i^{a_i}=\sum_{i=1}^n b_i^{b_i}\]

*Source: Own*

Hints can be found on my AoPS thread

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TopNewestBecause I love rephrasing problems:

Imagine an infinite set of weights, having the weights \(1^1, 2^2, 3^3, 4^4, \ldots\) units of weight, one of each. You have a two-pan balance. You want to put some number of weights on each pan such that the two pans balance. Prove that the only solution is to leave both pans empty.

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