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Suppose A

[tex] A_1 \cup A_2 \cup A_3 = \Delta[/tex]

and

[tex] F_i \cap A_i = \emptyset \text{ for } i=1,2,3[/tex]

Prove there exists some point [itex] x\in \Delta [/itex] such that

[tex] x\in A_1 \cap A_2 \cap A_3 [/tex]

Figurative bonus points if your solution involves a weakening or generalization of the hypothesis.

_{1}, A_{2}and A_{3}are closed convex sets, and let Δ be a triangle with edges F_{1}, F_{2}and F_{3}such that[tex] A_1 \cup A_2 \cup A_3 = \Delta[/tex]

and

[tex] F_i \cap A_i = \emptyset \text{ for } i=1,2,3[/tex]

Prove there exists some point [itex] x\in \Delta [/itex] such that

[tex] x\in A_1 \cap A_2 \cap A_3 [/tex]

Figurative bonus points if your solution involves a weakening or generalization of the hypothesis.

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