that simultaneously satisfy the conditions 
and![\[
\frac{x^4}{a} - \frac{y^4}{b} = \frac{1}{a - b}.
\]](http://latex.artofproblemsolving.com/4/5/e/45e29c956890cb0bd869c67e7ad94c92d868d2e2.png)
,![\[
\left( \frac{x^2}{a} \right)^n + \left( \frac{y^2}{b} \right)^n = \frac{2}{(a - b)^n}.
\]](http://latex.artofproblemsolving.com/4/0/5/405491f7aae19de4c7ad1f90da3d66e4038e3299.png)
and 
.
Equality holds when 
or 
or 
for all 
.
and 
, then we can deduce 
\le 2 $ ?
Y by
Adam and Bob are playing a game where they roll a fair 
-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 
.
-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 
.![\[
\mu = E\left[\frac{1}{2^a3^b}\right] = E\left[\frac{1}{2^a}\right]E\left[\frac{1}{3^b}\right]
\]](http://latex.artofproblemsolving.com/1/8/3/183178aea0a6209577fcc1ead3604afb7aad4d2c.png)
where 
and 
are uniformly distributed on 
.![\[
E\left[\frac{1}{2^a}\right] = \frac{1}{20}\sum_{a=1}^{20}\frac{1}{2^a}
=\frac{1}{20}\left(1-\frac{1}{2^{20}}\right)
\]](http://latex.artofproblemsolving.com/4/1/3/413efe66fba5a417dbd08b6a8b4a75adacc30020.png)
and![\[
E\left[\frac{1}{3^b}\right] = \frac{1}{20}\sum_{b=1}^{20}\frac{1}{3^b}
=\frac{1}{40}\left(1-\frac{1}{3^{20}}\right).
\]](http://latex.artofproblemsolving.com/6/0/c/60c79e3b10cad35b103e48031784cb87de361799.png)
So,![\[
\mu = \frac{1}{800}\left(1-\frac{1}{2^{20}}\right)\left(1-\frac{1}{3^{20}}\right).
\]](http://latex.artofproblemsolving.com/2/9/d/29d28ed96de0a41bcaf36a003fd23a1b24398e1b.png)
and now its easy to see that![\[
\left\lfloor\frac{1}{\mu}\right\rfloor = \left\lfloor 800\cdot\frac{1}{\left(1-\frac{1}{2^{20}}\right)\left(1-\frac{1}{3^{20}}\right)} \right\rfloor = 800.
\]](http://latex.artofproblemsolving.com/c/4/4/c4442c0b257d41fc311fa8cbd4d193896c382ed7.png)
