CN-LEAGUE 2005

In $\triangle ABC$ , $AB>AC$ , $l$ is a tangent line of the circumscribed circle of $\triangle ABC$ , passing through $A$ . The circle, centered at $A$ with radius $AC$ , intersects $AB$ at $D$ , and line $l$ at $E, F$ . Prove that lines $DE, DF$ pass through the incenter and an excenter of $\triangle ABC$ respectively.

Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$ , $az + cx = b$ , and $bx + ay = c$ . Find the minimum value of the function $$f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}.$$

For each positive integer, define a function $$f(n)=\begin{cases}0, &\text{if n is the square of an integer} \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}.$$ Find the value of $\sum_{k=1}^{200} f(k)$ .