KMO 2013

For positive integer $k$ , define integer sequence $\{ b_n \}, \{ c_n \}$ as follows: $$b_1 = c_1 = 1$$ $$b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1}$$ $$b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n}$$ Let $a_k = b_{2014}$ . Find the value of $$\sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }$$

For positive integer $a,b,c,d$ there are $a+b+c+d$ points on plane which none of three are collinear. Prove there exist two lines $l_1, l_2$ such that (1) $l_1, l_2$ are not parallel. (2) $l_1, l_2$ do not pass through any of $a+b+c+d$ points. (3) There are $a, b, c, d$ points on each region separated by two lines $l_1, l_2$ .