Geometric inequalities (Table of contents)
# Problems

**Problem 1**
**Problem 2**
**Problem 3**
**Problem 4**
**Problem 5**
**Problem 6**
**Problem 7**

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 \[MA\cdot MB\cdot MC\geq (MB_1+MC_1)\cdot(MC_1+MA_1)\cdot (MA_1+MB_1).\]

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

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

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 \[\frac{d_a}{h_a} +\frac{d_b}{h_b}+ \frac{d_c}{h_c}\geq\frac32.\]

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.

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?

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?