Competition problem — Algebra — Maths Olympiad Prep

Library / /72 of 860

Algebra Difficulty 4.7 AIME Find the answer

Let $N=2^{(2^{2})}$ and $x$ be a real number such that $N^{(N^{N})}=2^{(2^{x})}$ . Find $x$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

We compute $N^{(N^{N})}=16^{16^{16}}=2^{4 \cdot 2^{4 \cdot 2^{4}}}=2^{2^{2^{6}+2}}=2^{2^{66}}$ so $x=66$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.