Library / Philippine Mathematical Olympiad /14 of 114

Philippine Mathematical Olympiad

Algebra Difficulty 4.9 AIME Prove it Philippines

Problem:

Let $f(m) = 2^{2^{2 \cdots}}$ ( $m$ times). Find the least $m$ so that $\log_{10} f(m)$ exceeds $6$ .

Solution

Solution:

$f(4) = 2^{16} = 65536 < 10^{6} < 2^{65536} = f(5)$ .

Therefore, the least $m$ is $5$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.