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be positive real numbers such that 
. Prove that 
, denote by 
the number of positive divisors of . We say that a positive integer 
is Burapha integer if it satisfy the following condition
is an odd integer.
holds for every positive divisor 
of , such that 
be real numbers such that 
for all 
.
.
be a quadrilateral with 
. Diagonals 
and 
meet at 
. Let 
and 
denote the circumcircle and the circumcenter of triangle 
. Let 
and 
denote the circumcircle and circumcenter of triangle 
. Segment 
meets and again at 
and 
(other than 
and 
), respectively. Let 
and 
be the midpoints of minor arcs 
(not including ) and 
(not including ). Prove that 
.
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)
