Školjka

%V0 Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

%V0 Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Prove that $$a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0.$$ Determine when equality occurs.

%V0 Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

%V0 Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum $${BC\over PD}+{CA\over PE}+{AB\over PF}.$$

%V0 In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45\,^{\circ},$ $\angle BCP = \angle QCA = 30\,^{\circ},$ $\angle ABR = \angle RAB = 15\,^{\circ}.$ Prove that a.) $\angle QRP = 90\,^{\circ},$ and b.) $QR = RP.$

%V0 Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ $$a \cos^2{x}+b \cos{x}+c=0.$$ Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.