2004 AMC 12A Problems/Problem 16
Problem
The set of all real numbers for which
is defined is . What is the value of
?
Solution
For all real numbers and
such that
note that:
is defined if and only if
if and only if
Therefore, we have
from which
~Azjps (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Another solution
2001^a=x
2002^b = a
2003^c = b
2004^d = c
we can now rewrite the expression as x = 2001^(2002^(2003^(2004^d)))
the smallest value of x occurs when d approaches negative infinity. This makes the expression become
1.2001^(2002^(2003^0)) 2.2001^(2002^1)
which gives a final expression of 2001^2002
Video Solution (Logical Thinking)
~Education, the Study of Everything
See also
2004 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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 |