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inscribed in 
, 
are respectively the midpoints of the major and minor arcs 
. Let 
be 
, 
is the intersection point of the line through 
parallel to with . 
intersects 
at 
, take 
on 
so that 
. 
intersects at 
, 
intersects 
at 
. Prove that: 
is perpendicular to 
.
girls and boys are to be seated in a line. Find the total number of ways such that boys and girls are alternate and a particular boy and girl are never adjacent to each other in any arrangement.
players named 
is to be sent for batting. If wants to bat before 
and wants to bat after 
Then find the total number of batting orders if other players could go in any order.
that cannot be expressed as 
for some positive real number 
.
of cardboard, with 5 rectangular faces instead of 6 and there is no leftover cardboard. What is the largest volume this box can hold given that 
of cardboard is needed to cover the box?
can be written in the form![\[
\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},
\]](http://latex.artofproblemsolving.com/9/e/9/9e96c07ea3b0750af8bed344df6e8a86fcf0d772.png)
where 
and 
are positive integers. Find the value of 
.
, consider the generating function 
. Clearly, this yields 
. Therefore, we obtain
, we have
, it is straightforward to observe that the coefficient of 
in 
is 
. This leads to the conclusion.
, consider the generating function . Clearly, this yields . Therefore, we obtain, we have, it is straightforward to observe that the coefficient of in is . This leads to the conclusion.
and 
Prove that
be real numbers . Prove that
Subject to the constraints:
Prove that
. Prove that
and 
where each summation runs over 
and 
Then find the value of 
in 
is 
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