[/quote]
,
,![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
be the set of the first 
perfect squares of nonzero integers. Suppose that 
for some 
.
stands for the set 
.
holds for every 
.
such that 
,
satisfy that: 
satisfying 
is a perfect square
be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs 
such that 
.
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](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 
:
.
Y by
Given a circle 
and two points 
and 
outside , a quadrilateral 
is defined as "good" if 
are four distinct points on in order, and lines 
and 
intersect at and lines 
and 
intersect at .
For a quadrilateral
, let 
denote its area. If there exists a good quadrilateral, prove that there exists good quadrilateral such that for any good quadrilateral 
, 
.
and two points 
and 
outside , a quadrilateral 
is defined as "good" if 
are four distinct points on in order, and lines 
and 
intersect at and lines 
and 
intersect at .For a quadrilateral
, let 
denote its area. If there exists a good quadrilateral, prove that there exists good quadrilateral such that for any good quadrilateral 
, 
.This post has been edited 3 times. Last edited by steven_zhang123, Mar 29, 2025, 12:51 AM
.
wrt 
is fixed, and 
are polar conjugates, we may easily derive that 
.
and 
are harmonic, they form the same angles with ray 
,
and 
minus that of 
and 
,
.
and 
be the angle of line 
and 
from the long axis of 
respectively, then 
remains constant. Plugging in the equation in polar coordinate, we are curious about when 
achieves its maximum, where 
stands for the eccentricity. Some 
using the 
and 
both being constant. 
,
.
yields a fraction with numerator 
.
and 
,![$[-1, 1]$](http://latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
, thus uniquely determining the quadrilateral 
.
