Difference between revisions of "2013 Mock AIME I Problems/Problem 6"
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==Solution== | ==Solution== | ||
− | <math>f(x)=7\cos x+5\sin x</math> is a continuous function, so every value between its minimum and maximum is attainable. By Cauchy-Schwarz, <cmath>(7\cos x+5\sin x)^2 \le (7^2+5^2)(\cos^2 x+\sin^2 x)=74</cmath> Giving a maximum of <math>\sqrt{74}</math>, which is achievable when <math>\frac{\cos x}{7}=\frac{\sin x}{5}</math>. Note that a minimum of <math>-\sqrt{74}</math> can be attained at <math>f(x+\pi)</math>. Thus the values of <math>k</math> that work are the integers from <math>-4</math> to <math>3</math>, inclusive, giving a total of <math>\boxed{ | + | <math>f(x)=7\cos x+5\sin x</math> is a continuous function, so every value between its minimum and maximum is attainable. By Cauchy-Schwarz, <cmath>(7\cos x+5\sin x)^2 \le (7^2+5^2)(\cos^2 x+\sin^2 x)=74</cmath> Giving a maximum of <math>\sqrt{74}</math>, which is achievable when <math>\frac{\cos x}{7}=\frac{\sin x}{5}</math>. Note that a minimum of <math>-\sqrt{74}</math> can be attained at <math>f(x+\pi)</math>. Thus the values of <math>k</math> that work are the integers from <math>-4</math> to <math>3</math>, inclusive, giving a total of <math>\boxed{008}</math>. |
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+ | == See also == | ||
+ | * [[2013 Mock AIME I Problems]] | ||
+ | * [[2013 Mock AIME I Problems/Problem 5|Preceded by Problem 5]] | ||
+ | * [[2013 Mock AIME I Problems/Problem 7|Followed by Problem 7]] | ||
+ | |||
+ | [[Category:Intermediate Algebra Problems]] |
Revision as of 12:00, 30 July 2024
Problem 6
Find the number of integer values can have such that the equation
has a solution.
Solution
is a continuous function, so every value between its minimum and maximum is attainable. By Cauchy-Schwarz,
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
.