# 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?