VMO 2006

Solve the following system of equations in real numbers: $$\begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}.$$

Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$ , and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$ ). Prove that:

a) The point $N$ lies on a fixed circle;

b) The line $MN$ passes though a fixed point.

Let $m$ , $n$ be two positive integers greater than 3. Consider the table of size $m\times n$ ( $m$ rows and $n$ columns) formed with unit squares. We are putting marbles into unit squares of the table following the instructions:

$-$ each time put 4 marbles into 4 unit squares (1 marble per square) such that the 4 unit squares formes one of the followings 4 pictures (click here to view the pictures).

In each of the following cases, answer with justification to the following question: Is it possible that after a finite number of steps we can set the marbles into all of the unit squares such that the numbers of marbles in each unit square is the same?

a) $m=2004$ , $n=2006$ ;

b) $m=2005$ , $n=2006$ .

Attachments:

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$ , where $a$ , $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that $$f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} .$$

Find all polynomyals $P(x)$ with real coefficients which satisfy the following equality for all real numbers $x$ : $$P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 .$$

Let $S$ be a set of 2006 numbers. We call a subset $T$ of $S$ naughty if for any two arbitrary numbers $u$ , $v$ (not neccesary distinct) in $T$ , $u+v$ is not in $T$ . Prove that

1. If $S=\{1,2,\ldots,2006\}$ every naughty subset of $S$ has at most 1003 elements;

2. If $S$ is a set of 2006 arbitrary positive integers, there exists a naughty subset of $S$ which has 669 elements.