VMO 2006 — Division B

Solve the system
$$\begin{cases}x^3+3x^2+2x-5=y,\\ y^3+3y^2+2y-5=z,\\ z^3+3z^2+2z-5=x.\end{cases}$$

Let $ABCD$ be an isosceles trapezoid with $CD$ its larger base. A point $M$ moves on the line $CD$, distinct from $C,D$. Let $N$ be the second intersection, other than $M$, of circles $(BCM)$ and $(DAM)$. Prove that:
1. $N$ moves on a fixed circle.
2. The line $MN$ always passes through a fixed point.
Here $(XYZ)$ denotes the circle through $X,Y,Z$.

A forest ranger unit wants to prepare a patrol schedule for the whole year 2006 satisfying:
(i) The number of patrol days exceeds half the number of days in the year.
(ii) No two patrol days are exactly one week apart.
1. Prove that such a schedule exists, knowing that 2006 has $365$ days.
2. How many such patrol schedules are there?

Let an acute triangle $ABC$ be inscribed in a circle with center $O$, with $BC\gt AB\gt AC$. Line $OA$ meets line $BC$ at $A_1$, and line $OB$ meets line $CA$ at $B_2$. Let $B_1,C_1,C_2,A_2$ be the centers of circles $(AA_1B),(AA_1C),(BB_2C),(BB_2A)$, respectively. Prove that:
1. Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
2. These triangles are congruent if and only if angle $C$ of triangle $ABC$ equals $60^\circ$.
Here $(XYZ)$ denotes the circle through $X,Y,Z$.

Find all real-valued functions $f$ defined and continuous on $\mathbb R$ satisfying
$$f(x-y)f(y-z)f(z-x)+8=0$$
for all $x,y,z\in\mathbb R$.

Find the largest real number $k$ such that for all positive real numbers $a,b,c$ with $abc=1$, the inequality
$$\frac1{a^2}+\frac1{b^2}+\frac1{c^2}+3k\ge(k+1)(a+b+c)$$
always holds.