Question 126
Let $A=\left\lceil 1 / H_{3}\right\rceil$, $B=\left\lceil H_{5} / 2\right\rceil$. How many ways are there to partition the set $\{1,2, \ldots, A+B\}$ into two sets $U$ and $V$ with size $A$ and $B$ respectively such that the probability that a number chosen from $U$ uniformly at random is greater than a number chosen from $V$ uniformly at random is exactly $\frac{1}{2}$?