IMO Shortlist 1993 problem G3
Dodao/la:
arhiva2. travnja 2012. Let triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be such that its circumradius is
![R = 1.](/media/m/9/7/6/9761f58a9a377b899ecf20223bb4a366.png)
Let
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
be the inradius of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
and let
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
be the inradius of the orthic triangle
![A'B'C'](/media/m/5/3/d/53d1d147ad89bd52a7038ce57a0957ef.png)
of triangle
![ABC.](/media/m/c/b/7/cb77700b4adade65e440645391a8d2ad.png)
Prove that
%V0
Let triangle $ABC$ be such that its circumradius is $R = 1.$ Let $r$ be the inradius of $ABC$ and let $p$ be the inradius of the orthic triangle $A'B'C'$ of triangle $ABC.$ Prove that $$p \leq 1 - \frac{1}{3 \cdot (1+r)^2}.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1993