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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
2 years, 8 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. · 2 years, 8 months ago