Difference between revisions of "2019 AIME II Problems/Problem 6"
m (→Solution) |
|||
Line 27: | Line 27: | ||
<cmath>b^{36} = 6^{108}</cmath> | <cmath>b^{36} = 6^{108}</cmath> | ||
<cmath>b = 6^3 = \boxed{216}</cmath> | <cmath>b = 6^3 = \boxed{216}</cmath> | ||
+ | |||
+ | ==See Also== | ||
+ | {{AIME box|year=2019|n=II|num-b=5|num-a=7}} | ||
+ | {{MAA Notice}} |
Revision as of 17:50, 22 March 2019
Problem 6
In a Martian civilization, all logarithms whose bases are not specified as assumed to be base , for some fixed
. A Martian student writes down
and finds that this system of equations has a single real number solution
. Find
.
Solution
Using change of base on the second equation to base b,
Substituting this into the
of the first equation,
Using again,
However, since we found that ,
is also equal to
. Equating these,
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.