Problem 1. Which number is larger, $2^{5^{4^{4}}}$ or $3^{4^{2^{5}}}$?
Solution:

$2^{5^{4^{3}}} \vee 3^{4^{2^{5}}} \Leftrightarrow 2^{5^{64}} \vee 3^{4^{32}} \Leftrightarrow 2^{5^{64}}>2^{6^{64}}=2^{4^{63 \cdot 4}}=\left(2^{4}\right)^{4^{63}}=16^{4^{63}}>3^{4^{63}}>3^{4^{32}}$

Answer: $2^{5^{4^{3}}}>3^{4^{4^{5}}}$