Difference between revisions of "2016 AMC 12A Problems/Problem 20"
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We can manipulate the given identities to arrive at a conclusion about the binary operator <math>\diamondsuit</math>. Substituting <math>b = c</math> into the second identity yields <math>( a\ \diamondsuit\ b) \cdot b = a\ \diamondsuit\ (b\ \diamondsuit\ b) = a\ \diamondsuit\ 1 = a\ \diamondsuit\ ( a\ \diamondsuit\ a) = ( a\ \diamondsuit\ a) \cdot a = a</math>. Hence, <math>( a\ \diamondsuit\ b) \cdot b = a,</math> or, dividing both sides of the equation by <math>a,</math> <math>( a\ \diamondsuit\ b) = \frac{a}{b}.</math> | We can manipulate the given identities to arrive at a conclusion about the binary operator <math>\diamondsuit</math>. Substituting <math>b = c</math> into the second identity yields <math>( a\ \diamondsuit\ b) \cdot b = a\ \diamondsuit\ (b\ \diamondsuit\ b) = a\ \diamondsuit\ 1 = a\ \diamondsuit\ ( a\ \diamondsuit\ a) = ( a\ \diamondsuit\ a) \cdot a = a</math>. Hence, <math>( a\ \diamondsuit\ b) \cdot b = a,</math> or, dividing both sides of the equation by <math>a,</math> <math>( a\ \diamondsuit\ b) = \frac{a}{b}.</math> | ||
| − | Hence, the given equation becomes <math>\frac{2016}{\frac{ | + | Hence, the given equation becomes <math>\frac{2016}{\frac{6}{x}} = 100</math>. Solving yields <math>x=\frac{100}{336} = \frac{25}{84},</math> so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109.}</math> |
==See Also== | ==See Also== | ||
{{AMC12 box|year=2016|ab=A|num-b=19|num-a=21}} | {{AMC12 box|year=2016|ab=A|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 22:03, 4 February 2016
Problem
A binary operation 
has the properties that 
and that 
for all nonzero real numbers 
and 
(Here the dot 
represents the usual multiplication operation.) The solution to the equation 
can be written as 
where 
and 
are relativelt prime positive integers. What is 
Solution
We can manipulate the given identities to arrive at a conclusion about the binary operator . Substituting 
into the second identity yields 
. Hence, 
or, dividing both sides of the equation by 
Hence, the given equation becomes 
. Solving yields 
so the answer is 
See Also
| 2016 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 19 |
Followed by Problem 21 |
| 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. 
