CN-LEAGUE 2002

In $\triangle ABC$ , $\angle A = 60$ , $AB>AC$ , point $O$ is the circumcenter and $H$ is the intersection point of two altitudes $BE$ and $CF$ . Points $M$ and $N$ are on the line segments $BH$ and $HF$ respectively, and satisfy $BM=CN$ . Determine the value of $\frac{MH+NH}{OH}$ .

There are real numbers $a,b$ and $c$ and a positive number $\lambda$ such that $f(x)=x^3+ax^2+bx+c$ has three real roots $x_1, x_2$ and $x_3$ satisfying $(1) x_2-x_1=\lambda$ $(2) x_3>\frac{1}{2}(x_1+x_2)$ . Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$

Before The World Cup tournament, the football coach of $F$ country will let seven players, $A_1, A_2, \ldots, A_7$ , join three training matches (90 minutes each) in order to assess them. Suppose, at any moment during a match, one and only one of them enters the field, and the total time (which is measured in minutes) on the field for each one of $A_1, A_2, A_3$ , and $A_4$ is divisible by $7$ and the total time for each of $A_5, A_6$ , and $A_7$ is divisible by $13$ . If there is no restriction about the number of substitutions of players during each match, then how many possible cases are there within the total time for every player on the field?