CZ-SK-MO 2004

Find all triples $(x,y,z)$ of real numbers such that $$x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).$$

Consider all words containing only letters $A$ and $B$ . For any positive integer $n$ , $p(n)$ denotes the number of all $n$ -letter words without four consecutive $A$ 's or three consecutive $B$ 's. Find the value of the expression $$\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.$$

Given a circle $S$ and its $121$ chords $P_i (i=1,2,\ldots,121)$ , each with a point $A_i(i=1,2,\ldots,121)$ on it. Prove that there exists a point $X$ on the circumference of $S$ such that: there exist $29$ distinct indices $1\le k_1\le k_2\le\ldots\le k_{29}\le 121$ , such that the angle formed by ${A_{k_j}}X$ and ${P_{k_j}}$ is smaller than $21$ degrees for every $j=1,2,\ldots,29$ .

Find all positive integers $n$ such that $\sum_{k=1}^{n}\frac{n}{k!}$ is an integer.

Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$ . Let $K,M,N$ be the intersection points of $AL,CD$ ; $CL,AD$ ; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.

Find all functions $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that for all positive real numbers $x,y$ , $$x^2[f(x)+f(y)]=(x+y)f(yf(x)).$$