IMO Shortlist 1988 problem 13
Dodao/la:
arhiva2. travnja 2012. In a right-angled triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
let
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles
![ABD, ACD](/media/m/a/0/2/a0270bde62bb80bc735f13265d4563f0.png)
intersect the sides
![AB, AC](/media/m/d/7/6/d769eff805676f4c17bb0624e6a4ccef.png)
at the points
![K,L](/media/m/c/f/8/cf8b7b8c56970a06671ff82ddb7f6450.png)
respectively. If
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![E_1](/media/m/4/8/8/488028e5be310a0667dd7afafb1e6a96.png)
dnote the areas of triangles
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
and
![AKL](/media/m/d/6/d/d6d0d30ae4967203e1446dab0d3f5f18.png)
respectively, show that
%V0
In a right-angled triangle $ABC$ let $AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ABD, ACD$ intersect the sides $AB, AC$ at the points $K,L$ respectively. If $E$ and $E_1$ dnote the areas of triangles $ABC$ and $AKL$ respectively, show that
$$\frac {E}{E_1} \geq 2.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1988