IMO Shortlist 1993 problem G8
Dodao/la:
arhiva2. travnja 2012. The vertices
![D,E,F](/media/m/d/9/2/d9289350ce6acc48fa92a975875bc973.png)
of an equilateral triangle lie on the sides
![BC,CA,AB](/media/m/9/8/c/98c204ffa459114826231180fce7ec09.png)
respectively of a triangle
![ABC.](/media/m/c/b/7/cb77700b4adade65e440645391a8d2ad.png)
If
![a,b,c](/media/m/3/6/4/36454fdb50fc50f021324b33a6b513e3.png)
are the respective lengths of these sides, and
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
the area of
![ABC,](/media/m/8/a/f/8afcbd6e815ca10256c79a5b310e3d67.png)
prove that
%V0
The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC.$ If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC,$ prove that
$$DE \geq \frac{2 \cdot \sqrt{2} \cdot S}{\sqrt{a^2 + b^2 + c^2 + 4 \cdot \sqrt{3} \cdot S}}.$$
Izvor: Međunarodna matematička olimpijada, shortlist 1993