Difference between revisions of "2016 AMC 12A Problems/Problem 20"
(Created page with "==Problem 20== A binary operation <math>\diamondsuit </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\cdot c</math> and that...") |
(→Problem 20) |
||
Line 1: | Line 1: | ||
− | ==Problem | + | ==Problem== |
A binary operation <math>\diamondsuit </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\cdot c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> | A binary operation <math>\diamondsuit </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\cdot c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\ \cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relativelt prime positive integers. What is <math>p + q?</math> |
Revision as of 17:12, 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