Denote $\varrho$ and $p(\triangle DEF)$ the inradius and semiperimeter of $\triangle DEF.$ $R$ denotes the circumradius of $\triangle ABC$ and $X,Y,Z$ the feet of the altitudes on $BC,CA,AB.$ If $(u:v:w)$ represent the barycentric coordinates of $P$ WRT $\triangle ABC,$ then the area of its cevian triangle $\triangle DEF$ is given by $\frac{[DEF]}{[ABC]}=\frac{2uvw}{(v+w)(w+u)(u+v)}$ Since $P$ is inside $\triangle ABC,$ then $u,v,w >0.$ Thus by AM-GM we have $v+w \ge 2 \sqrt{vw} \ , \ w+u \ge 2 \sqrt{wu} \ , \ u+v \ge 2 \sqrt{uv} \Longrightarrow$ $(v+w)(w+u)(u+v) \ge 8 uvw \Longrightarrow \frac{1}{4} \ge \frac{2uvw}{(v+w)(w+u)(u+v)}=\frac{[DEF]}{[ABC]}.$ Since $\triangle ABC$ is acute, then its orthic triangle $\triangle XYZ$ is the inscribed triangle with the least perimeter (celebrated Fagnano's problem), for a proof see problem 4 in the thread Geometry problems. Hence $\ p(\triangle XYZ) \le p(\triangle DEF)$ together with the previous $\varrho \cdot p(\triangle DEF)=[DEF] \le \frac{_1}{^4}[ABC]$ gives $\varrho \le \frac{[ABC]}{4 \cdot p(\triangle XYZ)}=\frac{R}{4}.$

Where can I find the results of this National Olympiad?