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Problem 242

Algebra Difficulty 2.0 Prove it BMO Round 1 · United Kingdom · 2015

Place the following numbers in increasing order of size, and justify your reasoning:
$3^{3^4},\ 3^{4^3},\ 3^{4^4},\ 4^{3^3} \text{ and } 4^{3^4}.$
Note that $a^{b^c}$ means $a^{(b^c)}$ .

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; question papers published free at bmos.ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.