Geometric inequalities (Table of contents)

Problems

Problem 1

Let \(M\) be a point inside the triangle \(ABC\). Denote by \(A_1\), \(B_1\), and \(C_1\) the feet of perpendiculars from \(M\) to \(BC\), \(CA\), and \(AB\). Prove that \[MA\cdot MB\cdot MC\geq (MB_1+MC_1)\cdot(MC_1+MA_1)\cdot (MA_1+MB_1).\]

Problem 2

Prove that \[\frac1r=\frac1{r_a}+\frac1{r_b}+\frac1{r_c}=\frac1{h_a}+\frac1{h_b}+\frac1{h_c}\geq\frac1{l_a}+\frac1{l_b}+\frac1{l_c}\geq \frac1{m_a}+\frac1{m_b}+\frac1{m_c} \geq \frac2R.\]

Problem 3

Find all real numbers \(\alpha\) that satisfy: If \(a\), \(b\), \(c\) are sides of a triangle then \[a^2+b^2+c^2\leq \alpha(ab+bc+ca).\]

Problem 4

Let \(A_1\), \(B_1\), \(C_1\) be the intersections of the internal angle bisectors of the angles \(\angle A\), \(\angle B\), \(\angle C\) of \(\triangle ABC\). Denote by \(d_a\) the distance from \(A_1\) to the side \(AB\). Similarly we define \(d_b\) and \(d_c\). If \(h_a\), \(h_b\), and \(h_c\) are the lengths of the altitudes of \(\triangle ABC\) corresponding to the vertices \(A\), \(B\), and \(C\), prove that \[\frac{d_a}{h_a} +\frac{d_b}{h_b}+ \frac{d_c}{h_c}\geq\frac32.\]

Problem 5

A triangle \(ABC\) and three positive real numbers \(\alpha\), \(\beta\), and \(\gamma\) are given. Using a straight-edge and compass construct a point \(M\) in the plane for which \(\alpha MA+\beta MB+\gamma MC\) is minimal.

Problem 6

An equilateral triangle \(ABC\) is partitioned into \(n\) convex polygons. If no line intersects more than \(40\) of the given polygons, is it possible for \(n\) to be greater than a million?

Problem 7

Denote by \(h\) the longest altitude of the acute-angled triangle whose circumradius and inradius are \(R\) and \(r\) respectively. Prove that \[R+r\leq h.\] When does the equality hold?