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Friday, September 19, 2014

Powered distances

Let p and q be two infinitely long non-parallel lines on a plane. For any point X, let d_p(X), d_q(X) denote distances from X to p and q respectively. For any positive number r, let S(r) be the set of all points Y such that d_p(Y)^{2015} + d_q(Y)^{2015} \leq r.

Prove that S(r) is bounded.

Prove that S(r) is convex. That is, if A and B are in S(r) then AB is in S(r).

Identify all points in S such that d_p(X)^{2014} + d_q(X)^{2014} is maximum.

Prove that the area of S(r) is a convex function of r.

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