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and 
be positive real numbers such that 
. Find the minimum value of 
is acute triangle. 
is orthocenter, 
is the midpoint of 
, 
is the midpoint of smaller arc . Point 
is on 
such that, 
is perpendicular to 
. Prove that: 
is paralelogram
is a random variable reprasenting the number of girls in the teams combined. Clearly, 
.
is independent of 
, 
. and are hyper-geometric, that is
. Consider the ratio![\[\frac{\mathrm{P}(X_A = k+1)}{\mathrm{P}(X_A = k)} = 1 \implies k = \left\lfloor \frac{(m_2+1)(p+1)}{m_1+m_2+2} \right\rfloor\]](http://latex.artofproblemsolving.com/a/e/8/ae8dbf4a70f1768a3ca248770089b7a8bbd15cff.png)
has 
and otherway round for 
. to get the solution that 
is maximised at![\[
\boxed{
k = \left\lfloor \frac{(m_2+1)(p+1)}{m_1+m_2+2} \right\rfloor + \left\lfloor \frac{(n_2+1)(p+1)}{n_1+n_2+2} \right\rfloor
}
\]](http://latex.artofproblemsolving.com/a/3/4/a34bf0aad5c1305d2e1997825baf7af02cdfb47c.png)
and 
.
