Difference between revisions of "2022 AMC 12A Problems/Problem 17"
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==Problem== | ==Problem== | ||
| − | Supppose <math>a</math> is a real number such that the equation <cmath>a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}</cmath> has more than one solution in the interval <math>(0, \pi)</math>. The set of all such <math>a</math> that can be written | + | Supppose <math>a</math> is a real number such that the equation <cmath>a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}</cmath> |
| − | in the form <cmath>(p,q) \cup (q,r),</cmath> where <math>p, q,</math> and <math>r</math> are real numbers with <math>p < q< r</math>. What is <math>p+q+r</math>? | + | has more than one solution in the interval <math>(0, \pi)</math>. The set of all such <math>a</math> that can be written |
| + | in the form <cmath>(p,q) \cup (q,r),</cmath> | ||
| + | where <math>p, q,</math> and <math>r</math> are real numbers with <math>p < q< r</math>. What is <math>p+q+r</math>? | ||
<math>\textbf{(A) } -4 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 4</math> | <math>\textbf{(A) } -4 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 4</math> | ||
Revision as of 02:18, 12 November 2022
Problem
Supppose 
is a real number such that the equation ![\[a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}\]](http://latex.artofproblemsolving.com/6/5/1/6514e5c2f1b5f00055a46e13a0f7bc3a8b3ed885.png)
has more than one solution in the interval 
. The set of all such that can be written
in the form ![\[(p,q) \cup (q,r),\]](http://latex.artofproblemsolving.com/7/5/3/753fb68fe13dda58b70a340f28da137c751bdbf9.png)
where 
and 
are real numbers with 
. What is 
?
