KURSCHAK 1986

Let $n>2$ be a positive integer. Find the largest value $h$ and the smallest value $H$ for which $$h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H$$ holds for any positive reals $a_1,\dots,a_n$ .

A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$ ; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?