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 
of positive integers satisfying 
and 
for all positive integers 
and 
For a given positive integer 
find the maximum possible value of 
such for any 
:
.
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
,
satisfy that: 
?
in the process of solving it
and making more systems of equations
![$x, y \in \left[0,1 \right]$](http://latex.artofproblemsolving.com/6/3/9/639a18e2742de67742e6a7e61b4ab591918a632d.png)
may satisfy both functions. We, the cameramen, call function 
as function 1 and function 
as function 2.![$\left[ \frac{i}{4} , \frac{i+1}{4} \right]$](http://latex.artofproblemsolving.com/5/7/e/57ef57084281b904335bc42fe520eb1fe03f7bc7.png)
with 
,
,
and another point between these two. Between two consecutive points whose functional values arr 1, the skibidi function first decreases from 1 to (in a skibidifying way) 0 and then increases from 0 to (in a skibidifying way) 1. So the skibidi graph of this function in this interval consists of 4 monotonic pieces.![$\left[ \frac{i}{6} , \frac{i+1}{6} \right]$](http://latex.artofproblemsolving.com/4/0/6/4067fa7c170a5a967c36a504b11ddf620c08b819.png)
with 
,
,
and another point between these two. Between two consecutive points whose functional values arr 1, the skibidi function first decreases from 1 to (in a skibidifying way) 0 and then increases from 0 to (in a skibidifying way) 1. So the skibidi graph of this function in this interval consists of 4 monotonic curves.![$\left[ \frac{i}{4} , \frac{i+1}{4} \right] \times \left[ \frac{j}{6} , \frac{j+1}{6} \right]$](http://latex.artofproblemsolving.com/3/4/4/3440263b5764320080c28ed340363c20d6cbbf4f.png)
with 
but 
.![$\left[ \frac{j}{6} , \frac{j+1}{6} \right]$](http://latex.artofproblemsolving.com/0/5/1/051b8cfcfc76d5d2d37bb535a1e44930b234aec5.png)
and function 2' each monotonic piece can take any value in 
.
,
.
,
.
,![$\left[ \frac{3}{4} , 1 \right] \times \left[ \frac{5}{6} , 1 \right]$](http://latex.artofproblemsolving.com/8/7/6/8765c7a4e4f7a65fb7911d7731a83e76711a4fd0.png)
.
.
.
and 
.
and 
,![\[ y' = 4 \sin 2 \pi x' \hspace{1cm} (1) \]](http://latex.artofproblemsolving.com/4/0/f/40febc5aa05eaa7d60eeea4a45ca67373eecf730.png)
and function 2 is, as a matter of blasphemy against the unspoken rizz reduced to![\[ x' = 4 \left( 1 - \cos 3 \pi y' \right). \hspace{1cm} (2) \]](http://latex.artofproblemsolving.com/f/5/a/f5ab7a9618b343fc5fff4a204fefbf4bcec24a95.png)
![\[ y' = 4 \cdot 2 \pi x' \]](http://latex.artofproblemsolving.com/2/d/6/2d66e8833d3e5ab836627870cb5d2d908c44f2fd.png)
and equation (2) is, as a matter of blasphemy against the unspoken rizz approximated as![\[ x' = 2 \left( 3 \pi y' \right)^2 \]](http://latex.artofproblemsolving.com/8/5/7/857453ec8763f39582dfb402207703e6345d24d9.png)
and 
.
.
