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Omni-MATH / All subscripts in this problem are to be considered modulo 6 , that means for…

Problem

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problem

All subscripts in this problem are to be considered modulo 6 , that means for example that ω7\omega_{7} is the same as ω1\omega_{1}. Let ω1,…ω6\omega_{1}, \ldots \omega_{6} be circles of radius rr, whose centers lie on a regular hexagon of side length 1 . Let PiP_{i} be the intersection of ωi\omega_{i} and ωi+1\omega_{i+1} that lies further from the center of the hexagon, for i=1,…6i=1, \ldots 6. Let Qi,i=1…6Q_{i}, i=1 \ldots 6, lie on ωi\omega_{i} such that Qi,Pi,Qi+1Q_{i}, P_{i}, Q_{i+1} are colinear. Find the number of possible values of rr.
Plain-text mathematical notation (without MathML)
All subscripts in this problem are to be considered modulo 6 , that means for example that ω₇ is the same as ω₁. Let ω₁,…ω₆ be circles of radius r, whose centers lie on a regular hexagon of side length 1 . Let P_(i) be the intersection of ω_(i) and ω_(i+1) that lies further from the center of the hexagon, for i=1,…6. Let Q_(i),i=1…6, lie on ω_(i) such that Q_(i),P_(i),Q_(i+1) are colinear. Find the number of possible values of r.
Original LaTeX notation
All subscripts in this problem are to be considered modulo 6 , that means for example that $\omega_{7}$ is the same as $\omega_{1}$. Let $\omega_{1}, \ldots \omega_{6}$ be circles of radius $r$, whose centers lie on a regular hexagon of side length 1 . Let $P_{i}$ be the intersection of $\omega_{i}$ and $\omega_{i+1}$ that lies further from the center of the hexagon, for $i=1, \ldots 6$. Let $Q_{i}, i=1 \ldots 6$, lie on $\omega_{i}$ such that $Q_{i}, P_{i}, Q_{i+1}$ are colinear. Find the number of possible values of $r$.

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Source and history

task_id
77d15894-d527-54c8-9a42-cc10a24b3dfe
task_key
test--77d15894-d527-54c8-9a42-cc10a24b3dfe
task_revision_id
1
config / split
/ test

Official source

initial import