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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_17 / Suppose a is a real number such that the equation a⋅(sinx+sin(2x))=sin(3x) has more than one solution in the interval (0,π). The set of all such a that can be written in the form …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Suppose

a

is a real number such that the equation

a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}

has more than one solution in the interval

(0, \pi)

. The set of all such

a

that can be written in the form

(p,q) \cup (q,r),

where

p, q,

and

r

are real numbers with

p < q< r

. What is

p+q+r

Plain-text mathematical notation (without MathML)

Suppose a is a real number such that the equation a⋅(sinx+sin(2x))=sin(3x)
has more than one solution in the interval (0,π). The set of all such a that can be written
in the form (p,q)∪(q,r),
where p,q, and r are real numbers with p<q<r. What is p+q+r?

Original LaTeX notation

Suppose $a$ is a real number such that the equation \[a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)}\]
has more than one solution in the interval $(0, \pi)$. The set of all such $a$ that can be written
in the form \[(p,q) \cup (q,r),\]
where $p, q,$ and $r$ are real numbers with $p < q< r$. What is $p+q+r$?

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_17
task_id: 645533d4-4b22-59ca-8ebe-9259b727a057
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2022~5fAMC~5f12A~5fProblems~2fProblem~5f17
task_revision_id: 2
config / split: default / train

Official source

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