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Omni-MATH / Regular hexagon P₁P₂P₃P₄P₅P₆ has side length 2. For 1≤i≤6, let C_(i) be a unit circle centered at P_(i) and ℓ_(i) be one of the internal common tangents of C_(i) and C_(i+2), where…

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problem

Regular hexagon P1P2P3P4P5P6P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} has side length 2. For 1≤i≤61 \leq i \leq 6, let CiC_{i} be a unit circle centered at PiP_{i} and ℓi\ell_{i} be one of the internal common tangents of CiC_{i} and Ci+2C_{i+2}, where C7=C1C_{7}=C_{1} and C8=C2C_{8}=C_{2}. Assume that the lines {ℓ1,ℓ2,ℓ3,ℓ4,ℓ5,ℓ6}\{\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}, \ell_{5}, \ell_{6}\} bound a regular hexagon. The area of this hexagon can be expressed as ab\sqrt{\frac{a}{b}}, where aa and bb are relatively prime positive integers. Compute 100a+b100 a+b.
Plain-text mathematical notation (without MathML)
Regular hexagon P₁P₂P₃P₄P₅P₆ has side length 2. For 1≤i≤6, let C_(i) be a unit circle centered at P_(i) and ℓ_(i) be one of the internal common tangents of C_(i) and C_(i+2), where C₇=C₁ and C₈=C₂. Assume that the lines {ℓ₁,ℓ₂,ℓ₃,ℓ₄,ℓ₅,ℓ₆} bound a regular hexagon. The area of this hexagon can be expressed as √((a)/(b)), where a and b are relatively prime positive integers. Compute 100a+b.
Original LaTeX notation
Regular hexagon $P_{1} P_{2} P_{3} P_{4} P_{5} P_{6}$ has side length 2. For $1 \leq i \leq 6$, let $C_{i}$ be a unit circle centered at $P_{i}$ and $\ell_{i}$ be one of the internal common tangents of $C_{i}$ and $C_{i+2}$, where $C_{7}=C_{1}$ and $C_{8}=C_{2}$. Assume that the lines $\{\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}, \ell_{5}, \ell_{6}\}$ bound a regular hexagon. The area of this hexagon can be expressed as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

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Source and history

task_id
e6f54e41-a915-540d-9dbf-5ec08155fc5b
task_key
test--e6f54e41-a915-540d-9dbf-5ec08155fc5b
task_revision_id
1
config / split
/ test

Official source

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