2019 AMC 12A Problems/Problem 23
Problem
Define binary operations and
by
for all real numbers
and
for which these expressions are defined. The sequence
is defined recursively by
and
for all integers
. To the nearest integer, what is
?
Solution
Using the recursive definition, or
where
and
. Using logarithm rules, we can remove the exponent of the 3 so that
. Therefore,
, which is
.
We claim that for all
. We can prove this through induction.
See Also
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.