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Omni-MATH / Let ABC be an acute triangle with A-excircle Γ. Let the line through A perpendicular to BC intersect BC at D and intersect Γ at E and F. Suppose that AD=DE=EF. If the maximum value…

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problem

Let ABCA B C be an acute triangle with AA-excircle Γ\Gamma. Let the line through AA perpendicular to BCB C intersect BCB C at DD and intersect Γ\Gamma at EE and FF. Suppose that AD=DE=EFA D=D E=E F. If the maximum value of sinB\sin B can be expressed as a+bc\frac{\sqrt{a}+\sqrt{b}}{c} for positive integers a,ba, b, and cc, compute the minimum possible value of a+b+ca+b+c.
Plain-text mathematical notation (without MathML)
Let ABC be an acute triangle with A-excircle Γ. Let the line through A perpendicular to BC intersect BC at D and intersect Γ at E and F. Suppose that AD=DE=EF. If the maximum value of sinB can be expressed as (√(a)+√(b))/(c) for positive integers a,b, and c, compute the minimum possible value of a+b+c.
Original LaTeX notation
Let $A B C$ be an acute triangle with $A$-excircle $\Gamma$. Let the line through $A$ perpendicular to $B C$ intersect $B C$ at $D$ and intersect $\Gamma$ at $E$ and $F$. Suppose that $A D=D E=E F$. If the maximum value of $\sin B$ can be expressed as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, compute the minimum possible value of $a+b+c$.

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task_id
fb6d56b4-7444-52a3-a553-e3a1551ef883
task_key
test--fb6d56b4-7444-52a3-a553-e3a1551ef883
task_revision_id
1
config / split
/ test

Official source

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