Problem

Source: Romanian TST 3 2007, Problem 4

Tags: Euler, inequalities, geometry, circumcircle, geometric transformation, homothety, cyclic quadrilateral



The points $M, N, P$ are chosen on the sides $BC, CA, AB$ of a triangle $\Delta ABC$, such that the triangle $\Delta MNP$ is acute-angled. We denote with $x$ the length of the shortest altitude of the triangle $\Delta ABC$, and with $X$ the length of the longest altitudes of the triangle $\Delta MNP$. Prove that $x \leq 2X$.