AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
. Prove that the graph of is symmetric about a point (i.e., it has a center of symmetry).![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
![\[
|MA|^2 + |MB|^2 + |MC|^2 = |GA|^2 + |GB|^2 + |GC|^2 + 3|MG|^2
\]](http://latex.artofproblemsolving.com/0/7/0/070ff15920d19d474defb77ceb606b2811fa85b4.png)
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](http://latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
be a positive root of the equation 
. Prove that the following sequence converges: 
of positive integers satisfying 
and 
for all positive integers 
and 
For a given positive integer 
find the maximum possible value of 
such for any 
:
.
prove that
Hence, as a corollary, show
be a positive root of the equation 
. Prove that the following sequence converges: 
, just don’t know how to prove it.
,
satisfy that: 
?
hence ![${{n}^{2}}(\sqrt[n]{a}-\sqrt[n+1]{a})={{n}^{2}}\sqrt[n+1]{a}\left( \frac{\sqrt[n]{a}}{\sqrt[n+1]{a}}-1 \right)={{n}^{2}}\sqrt[n+1]{a}\left( {{a}^{\frac{1}{n(n+1)}}}-1 \right)=$](http://latex.artofproblemsolving.com/d/f/e/dfe9afef2115ded44a29597baa9c9884a9b8f3e6.png)
![$=\frac{n}{n+1}\sqrt[n+1]{a}\frac{{{a}^{\frac{1}{n(n+1)}}}-1}{\frac{1}{n(n+1)}}\to \ln a=\ln ({{x}^{2}}+x+1)$](http://latex.artofproblemsolving.com/e/f/6/ef670bda07317faa7fd305ee1a089ef8d71ab387.png)
