Question 5
Let $(O)$ be a circle with diameter $AB$, and let $M$ be any point strictly inside the circle, not on $AB$. The internal angle-bisector ray at $M$ of triangle $AMB$ meets $(O)$ at $N$. The external angle bisector of $\angle AMB$ meets the lines $NA,NB$ at $P,Q$, respectively. The line $MA$ meets the circle with diameter $NQ$ again at $R$, and the line $MB$ meets the circle with diameter $NP$ again at $S$, where $R,S\ne M$. Prove that the median from $N$ of triangle $NRS$ always passes through a fixed point as $M$ varies inside the circle.
Source defect: for some permitted interior points M, a specified second intersection is tangent at M and cannot satisfy R,S different from M. The intended theorem holds wherever the construction is defined; alternatively allow a repeated intersection at tangency. The solutions give an explicit counterexample to the unrestricted construction and prove the corrected interpretation.
Source defect: for some permitted interior points M, a specified second intersection is tangent at M and cannot satisfy R,S different from M. The intended theorem holds wherever the construction is defined; alternatively allow a repeated intersection at tangency. The solutions give an explicit counterexample to the unrestricted construction and prove the corrected interpretation.