is 
.
,
.
.
that satisfies![\[
a^3+b^3=1957
\]](http://latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](http://latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
Prove that
Let 
Prove that
be the solid square region with vertices at![\[
(0, 0), \left(\frac{1}{2}, \frac{1}{2}\right), (1, 0), \left(\frac{1}{2}, -\frac{1}{2}\right).
\]](http://latex.artofproblemsolving.com/a/9/8/a9898f9237f07eaef5d17a9ac689d80ca631863f.png)
consists of two copies of 
:
counterclockwise around the origin and scaled by a factor of 
,
.
below.
be the limiting region of the sequence 
.
for relatively prime positive integers 
and 
.
.
be prime numbers such that![\[
\frac{1}{pq}+\frac{1}{qr}+\frac{1}{pr}=\frac{1}{839}
\]](http://latex.artofproblemsolving.com/b/6/f/b6f882f3c93b0877419db625c2a427fc9561605b.png)
Hence find the numbers of ordered triples 
(i.e., the square on 
has 
,
,
to the centers of the opposite squares are concurrent.
be the roots of the equation 
.
if![\[
Z=\frac{\alpha+\beta+\gamma+\delta}{\alpha(\beta+\gamma+\delta)+\beta(\gamma+\delta)}+\alpha\beta\gamma\delta[\alpha\beta(\gamma+\delta)+\gamma\delta(\alpha+\beta)]
\]](http://latex.artofproblemsolving.com/4/2/5/425d9100db9e90264b50fc7c4b062ddb53c64068.png)
denote the greatest integer less than or equal to 
.
,![\[
M = \left\lfloor \frac{1^2}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor + \left\lfloor \frac{3^2}{3} \right\rfloor + \dots + \left\lfloor \frac{39^2}{3} \right\rfloor + \left\lfloor \frac{40^2}{3} \right\rfloor
\]](http://latex.artofproblemsolving.com/1/d/4/1d4576c5f14b3f43a99f273eb2e8133388945135.png)
Y by
Howdy!
I'm in Calculus 3556. I remember seeing something that said if I email someone, I can ask for longer access to the course page? I would like to have the materials from class up until I take the AP Exam in mid May. I'm assuming I would need to email Student Services? Would that be studentservices@aops.com?
Thanks!
I'm in Calculus 3556. I remember seeing something that said if I email someone, I can ask for longer access to the course page? I would like to have the materials from class up until I take the AP Exam in mid May. I'm assuming I would need to email Student Services? Would that be studentservices@aops.com?
Thanks!
