DE-MO 2004

Find all real numbers $x,y$ satisfying the following system of equations \begin{align*} x^4 +y^4 & =17(x+y)^2 xy & =2(x+y). \end{align*}

Let $k$ be a circle with center $M.$ There is another circle $k_1$ whose center $M_1$ lies on $k,$ and we denote the line through $M$ and $M_1$ by $g.$ Let $T$ be a point on $k_1$ and inside $k.$ The tangent $t$ to $k_1$ at $T$ intersects $k$ in two points $A$ and $B.$ Denote the tangents (diifferent from $t$ ) to $k_1$ passing through $A$ and $B$ by $a$ and $b$ , respectively. Prove that the lines $a,b,$ and $g$ are either concurrent or parallel.

Prove that for every positive integer $n$ there is an $n$ -digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.

For a positive integer $n,$ let $a_n$ be the integer closest to $\sqrt{n}.$ Compute $$\frac{1}{a_1 } + \frac{1}{a_2 }+ \cdots + \frac{1}{a_{2004}}.$$

Prove that for four positive real numbers $a,b,c,d$ the following inequality holds and find all equality cases: $$a^3 +b^3 +c^3 +d^3 \geq a^2 b +b^2 c+ c^2 d +d^2 a.$$