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Prove equality never occurs

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

Note by Daniel Liu
3 years, 7 months ago

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Because 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.

Ivan Koswara - 3 years, 7 months ago

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