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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