Difference between revisions of "2013 Mock AIME I Problems/Problem 6"
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Revision as of 00:27, 5 March 2017
Problem 6
Find the number of integer values 
can have such that the equation ![\[7\cos x+5\sin x=2k+1\]](http://latex.artofproblemsolving.com/4/3/a/43a0679c9612e94f6b3c566f42ce670f09a1caa7.png)
has a solution.
Solution
is a continuous function, so every value between its minimum and maximum is attainable. By Cauchy-Schwarz, ![\[(7\cos x+5\sin x)^2 \ge (7^2+5^2)(\cos^2 x+\sin^2 x)=74\]](http://latex.artofproblemsolving.com/2/7/7/277395215874a2a1e9b76fdaba3c590bf0386607.png)
Giving a maximum of 
, which is achievable when 
. Note that a minimum of 
can be attained at 
. Thus the values of that work are the integers from 
to 
, inclusive, giving a total of 
.
