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MATH / Let α, β, and γ be three real numbers. Suppose that

Problem

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problem

Let α,\alpha, β,\beta, and γ\gamma be three real numbers. Suppose that \begin{align*} \cos \alpha + \cos \beta + \cos \gamma &= 1, \\ \sin \alpha + \sin \beta + \sin \gamma &= 1. \end{align*}Then the minimum value of cosα\cos \alpha can be expressed in the form −p+qr,-\frac{p + \sqrt{q}}{r}, where p,p, q,q, and rr are positive integers, and qq is not divisible by the square of a prime. Find p+q+r.p + q + r.
Plain-text mathematical notation (without MathML)
Let α, β, and γ be three real numbers.  Suppose that
\begin{align*}
\cos \alpha + \cos \beta + \cos \gamma &= 1, \\
\sin \alpha + \sin \beta + \sin \gamma &= 1.
\end{align*}Then the minimum value of cosα can be expressed in the form −(p+√(q))/(r), where p, q, and r are positive integers, and q is not divisible by the square of a prime.  Find p+q+r.
Original LaTeX notation
Let $\alpha,$ $\beta,$ and $\gamma$ be three real numbers.  Suppose that
\begin{align*}
\cos \alpha + \cos \beta + \cos \gamma &= 1, \\
\sin \alpha + \sin \beta + \sin \gamma &= 1.
\end{align*}Then the minimum value of $\cos \alpha$ can be expressed in the form $-\frac{p + \sqrt{q}}{r},$ where $p,$ $q,$ and $r$ are positive integers, and $q$ is not divisible by the square of a prime.  Find $p + q + r.$

level

Level 5

type

Precalculus

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Source and history

task_id
75732aab-e66b-5c49-87a1-bffda352735f
task_key
precalculus--test--75732aab-e66b-5c49-87a1-bffda352735f
task_revision_id
1
config / split
precalculus / test

Official source

initial import