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 triangle whose circumradius and inradius are \( R \) and \( r \) respectively. Prove that \[ R+r\leq h.\] When does the equality hold?


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