Školjka

%V0 Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality $$abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.$$ Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

%V0 Consider two segments of length $a, b \ (a > b)$ and a segment of length $c = \sqrt{ab}$. (a) For what values of $a/b$ can these segments be sides of a triangle ? (b) For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled ?

%V0 Given two congruent triangles $A_1A_2A_3$ and $B_1B_2B_3$ ($A_iA_k = B_iB_k$), prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented.

%V0 Given a point $O$ and lengths $x, y, z$, prove that there exists an equilateral triangle $ABC$ for which $OA = x, OB = y, OC = z$, if and only if $x+y \geq z, y+z \geq x, z+x \geq y$ (the points $O,A,B,C$ are coplanar).

%V0 If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB', CC'$ pass through a given point.

%V0 Given $n \ (n \geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2, \ldots,A_n$ in such a way that for each $i, j, k, 1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ .$