AoPS Academy
that satises 
and 
for real 
and 
can be written as 
where the greatest common divisor of positive integers 
, 
, and 
is 
, and 
is not divisible by the square of any prime. Compute the value of 
.
-sided die with faces labeled with the integers through , obtaining and , respectively. They then calculate their cumulative score for the game, given by the value of 
. Let 
denote the expected value of this score. Compute the value of the greatest integer less than or equal to 
.
has side lengths 
and 
. Consider a point 
on 
. 
is rotated about 
to 
so that 
, and 
lie on the same line and 
is perpendicular to 
. If 
, where 
and 
are relatively prime positive integers, find 
.
be positive numbers such that 
. Find the minimum value of:
2, Let be positive numbers such that 
. Find the maximum value of:
and 
respectively.And then we verify 
in term of 
and 
. 
define 
on ![$y \in [0,\sqrt{2}] $](http://latex.artofproblemsolving.com/c/c/3/cc350547384b66e783e5326642712eba395f91d5.png)
![$\frac{a}{c}+\frac{b}{c}+\frac{c^2}{ab} \ge 3\sqrt[3]{\frac{a}{c}\cdot\frac{b}{c}\cdot\frac{c^2}{2\sqrt{2}ab}}+\frac{(4-\sqrt{2})c^2}{4ab} \ge 3\sqrt[3]{\frac{1}{2\sqrt{2}}}+\frac{(4-\sqrt{2})2ab}{4ab}=2+\sqrt{2}$](http://latex.artofproblemsolving.com/4/6/f/46f3aae4f14731a6d8196d517c702b009cf4aa6a.png)
.
) that 
. Suppose, that 
. Then 
. So 
and 
. When 
the prove will be finished, but now we have to figure out what if 
?
is decreasing for 
, so if 
, then 
.
, then read two first sentences from this post., then as in post above , so 
. Adding to both sides 
gives 
.
. So here 
with equality iff 
.
where 
are positive, using Cauchy-Schwarz, we have:
) that . Suppose, that . Then . So and . When the prove will be finished, but now we have to figure out what if ?
then prove 
![$\frac{\frac{a}{c}+\frac{b}{c}+\frac{c^2}{ab}}{3} \ge \sqrt[3]{\frac{a}{c}\cdot\frac{b}{c}\cdot\frac{c^2}{ab}}=1$](http://latex.artofproblemsolving.com/8/e/f/8ef0cba1041039f2a295069bba2d01388b610da3.png)
.
which is a contradiction. So
are the sides of rectangular triangle.
which holds by triangle inequality.
and 
.
,
we have 
.