I would appreciate any help in solving this question:
Students in one classroom answer correctly with probability greater than 0.5. Students in the other classroom answer correctly with probability less than 0.5. Teacher answers correctly with probability greater than 0.5. Together they answer correctly with probability of exactly 0.5. What is the ratio of students in those two classrooms?
If there are $s_1$ students in the first class with a probability of $p_1$ each of being correct, and $s_2$ students in the second class with a probability of $p_2$ each of being correct, and the teacher has a probability of $p_t$ of being correct, then "Together they answer correctly with probability of exactly $0.5$" suggests to me that any positive rational ratio of $s_1:s_2$ is possible.
For example for the ratio $a:b$ consider $$s_1=a, s_2=b, \; p_1=p_t=\frac12 +\frac{b}{2(a+b+1)} \gt \frac12, \;p_2= \frac{b}{2(a+b+1)} \lt \frac12.$$
This has $$\frac{\displaystyle a \left(\frac12 +\frac{b}{2(a+b+1)}\right) + b \left(\frac{b}{2(a+b+1)}\right) +\left(\frac12 +\frac{b}{2(a+b+1)}\right)}{a+b+1} =\frac{1}2.$$