IMO Shortlist 1967 problem 4
Dodao/la:
arhiva2. travnja 2012. Suppose medians
![m_a](/media/m/7/0/f/70f036394b3e46bd4862186d92f1b22f.png)
and
![m_b](/media/m/0/f/f/0ffffc68293f0dd95bab1e4dfac7ba36.png)
of a triangle are orthogonal. Prove that:
a.) Using medians of that triangle it is possible to construct a rectangular triangle.
b.) The following inequality:
![5(a^2+b^2-c^2) \geq 8ab,](/media/m/9/3/4/9346ee2be06120a9086741cc2ff45e2a.png)
is valid, where
![a,b](/media/m/7/d/8/7d8bdace47e602448e6040957d8cf923.png)
and
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
are side length of the given triangle.
%V0
Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that:
a.) Using medians of that triangle it is possible to construct a rectangular triangle.
b.) The following inequality: $$5(a^2+b^2-c^2) \geq 8ab,$$ is valid, where $a,b$ and $c$ are side length of the given triangle.
Izvor: Međunarodna matematička olimpijada, shortlist 1967