Difference between revisions of "2021 AMC 12A Problems/Problem 18"
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==Problem== | ==Problem== | ||
| − | + | Let <math>f</math> be a function defined on the set of positive rational numbers with the property that <math>f(a\cdot b) = f(a)+f(b)</math> for all positive rational numbers <math>a</math> and <math>b</math>. Furthermore, suppose that <math>f</math> also has the property that <math>f(p)=p</math> for every prime number <math>p</math>. For which of the following values <math>x</math> is <math>f(x) < 0</math>? | |
| − | + | ||
| − | + | <math>\textbf{(A) }\frac{17}{32}\qquad\textbf{(B) }\frac{11}{16}\qquad\textbf{(C) }\frac{7}{9}\qquad\textbf{(D) }\frac{7}{6}\qquad\textbf{(E) }\frac{25}{11}\qquad</math> | |
| − | == | + | |
| − | + | ==Video Solution by Punxsutawney Phil== | |
| + | https://youtu.be/8gGcj95rlWY | ||
| + | |||
==See also== | ==See also== | ||
{{AMC12 box|year=2021|ab=A|num-b=17|num-a=19}} | {{AMC12 box|year=2021|ab=A|num-b=17|num-a=19}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 14:16, 11 February 2021
Problem
Let 
be a function defined on the set of positive rational numbers with the property that 
for all positive rational numbers 
and 
. Furthermore, suppose that also has the property that 
for every prime number 
. For which of the following values 
is 
?
Video Solution by Punxsutawney Phil
See also
| 2021 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 17 |
Followed by Problem 19 |
| 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. 
