Difference between revisions of "2016 AMC 12A Problems/Problem 16"
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<math>y = -\frac{1}{t}</math> | <math>y = -\frac{1}{t}</math> | ||
− | where <math>t = \log_3 x</math>. | + | where <math>t = \log_3 x</math>. Note that all values of <math>t</math> correspond to exactly one positive <math>x</math> value, so all <math>(t,y)</math> intersections will correspond to exactly one <math>(x,y)</math> intersection in the positive-x area. |
− | Graphing this system of | + | Graphing this system of functions will generate a total of <math>5</math> solutions <math>\rightarrow \boxed{\textbf{(D)} 5}</math> |
==See Also== | ==See Also== | ||
{{AMC12 box|year=2016|ab=A|num-b=15|num-a=17}} | {{AMC12 box|year=2016|ab=A|num-b=15|num-a=17}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:05, 19 July 2021
Contents
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
Solution 2
Note that .
Then
Therefore, the system of equations can be simplified to:
where . Note that all values of correspond to exactly one positive value, so all intersections will correspond to exactly one intersection in the positive-x area.
Graphing this system of functions will generate a total of solutions
See Also
2016 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 |
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