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Omni-MATH / A regular hexagon ABCDEF has side length 1 and center O. Parabolas P₁,P₂,…,P₆ are constructed with common focus O and directrices AB,BC,CD,DE,EF,FA respectively. Let χ be the set o…

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problem

A regular hexagon ABCDEFA B C D E F has side length 1 and center OO. Parabolas P1,P2,…,P6P_{1}, P_{2}, \ldots, P_{6} are constructed with common focus OO and directrices AB,BC,CD,DE,EF,FAA B, B C, C D, D E, E F, F A respectively. Let χ\chi be the set of all distinct points on the plane that lie on at least two of the six parabolas. Compute ∑X∈χ|OX|\sum_{X \in \chi}|O X| (Recall that the focus is the point and the directrix is the line such that the parabola is the locus of points that are equidistant from the focus and the directrix.)
Plain-text mathematical notation (without MathML)
A regular hexagon ABCDEF has side length 1 and center O. Parabolas P₁,P₂,…,P₆ are constructed with common focus O and directrices AB,BC,CD,DE,EF,FA respectively. Let χ be the set of all distinct points on the plane that lie on at least two of the six parabolas. Compute ∑_(X∈χ)|OX| (Recall that the focus is the point and the directrix is the line such that the parabola is the locus of points that are equidistant from the focus and the directrix.)
Original LaTeX notation
A regular hexagon $A B C D E F$ has side length 1 and center $O$. Parabolas $P_{1}, P_{2}, \ldots, P_{6}$ are constructed with common focus $O$ and directrices $A B, B C, C D, D E, E F, F A$ respectively. Let $\chi$ be the set of all distinct points on the plane that lie on at least two of the six parabolas. Compute $$\sum_{X \in \chi}|O X|$$ (Recall that the focus is the point and the directrix is the line such that the parabola is the locus of points that are equidistant from the focus and the directrix.)

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task_id
44652cd8-0a00-5ac1-878b-6b7b2a941286
task_key
test--44652cd8-0a00-5ac1-878b-6b7b2a941286
task_revision_id
1
config / split
/ test

Official source

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