tmua.me Tau Paper 2 · Question 12

An equilateral triangle $ABC$ has circumcentre $O$ and circumradius $1$ . For $r>0$ , let $\mathcal R_r$ be the set of points $P$ on the circle $OP=r$ for which the three numbers

PA^2,\quad PB^2,\quad PC^2

are the side lengths of a non-degenerate right-angled triangle. Let $\Delta(P)$ be the area of that right-angled triangle.

Consider only radii $r$ for which $\mathcal R_r$ contains exactly six distinct points, and define

E(r)=\frac{\displaystyle\sum_{P\in\mathcal R_r}\Delta(P)}{(1+r^2)^2}.

Which interval is exactly the set of possible values of $E(r)$ ?

\displaystyle \left(0,\frac{81-54\sqrt2}{2}\right]

\displaystyle \left(0,\frac{81-54\sqrt2}{4}\right)

\displaystyle \left(0,\frac{81-54\sqrt2}{2}\right)

\displaystyle \left[0,\frac{81-54\sqrt2}{2}\right)

\displaystyle \left[0,\frac{81-54\sqrt2}{2}\right]

examday

tmua.me Tau Paper 2 · Q12