CN-LEAGUE 2004

In an acute triangle $ABC$ , point $H$ is the intersection point of altitude $CE$ to $AB$ and altitude $BD$ to $AC$ . A circle with $DE$ as its diameter intersects $AB$ and $AC$ at $F$ and $G$ , respectively. $FG$ and $AH$ intersect at point $K$ . If $BC=25$ , $BD=20$ , and $BE=7$ , find the length of $AK$ .

In a planar rectangular coordinate system, a sequence of points ${A_n}$ on the positive half of the y-axis and a sequence of points ${B_n}$ on the curve $y=\sqrt{2x}$ $(x\ge0)$ satisfy the condition $|OA_n|=|OB_n|=\frac{1}{n}$ . The x-intercept of line $A_nB_n$ is $a_n$ , and the x-coordinate of point $B_n$ is $b_n$ , $n\in\mathbb{N}$ . Prove that (1) $a_n>a_{n+1}>4$ , $n\in\mathbb{N}$ ; (2) There is $n_0\in\mathbb{N}$ , such that for any $n>n_0$ , $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\ldots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004$ .

For integer $n\ge 4$ , find the minimal integer $f(n)$ , such that for any positive integer $m$ , in any subset with $f(n)$ elements of the set ${m, m+1, \ldots, m+n+1}$ there are at least $3$ relatively prime elements.