Difference between revisions of "2016 AMC 12A Problems/Problem 16"
(Created page with "==Problem 16== The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many p...") |
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Revision as of 17:27, 4 February 2016
Problem 16
The graphs of and
are plotted on the same set of axes. How many points in the plane with positive
-coordinates lie on two or more of the graphs?
Solution
Setting the first two equations equal to each other, .
Solving this, we get and
.
Similarly with the last two equations, we get and
.
Now, by setting the first and third equations equal to each other, we get .
Pairing the first and fourth or second and third equations won't work because then .
Pairing the second and fourth equations will yield , but since you can't divide by
, it doesn't work.
After trying all pairs, we have a total of solutions