be a polynomial with integer coefficients such that 
and 
for some integer 
.
such that 
.
and 
Then 
in terms of 
for some real number 
and ![\[ a_n=na_{n-1}+(n-1)(n!(n-1)!-1) \]](http://latex.artofproblemsolving.com/4/1/f/41f0cc7b041a8bba056882b5c794503195c374fd.png)
for integers 
.
,
,
and 
are positive integers whose greatest common divisor is 
,
and 
are in the coordinate plane such that 
.
denote the locus of all points 
in the coordinate plane satisfying 
,
be the midpoint of 
.
and 
are on 
and 
.
can be expressed in the form 
,
and 
are relatively prime positive integers. Find 
.
Prove that
Let 
Prove that
satisfying:
Y by Acorndoorbell
Summary of the problem: Trying dot((0,0)) on asymptote breaks it
Page URL: Any old AoPS page
Steps to reproduce:
1. Open up an asymptote window on AoPS
2. Type dot((0,0));
Expected behavior: It should show you a dot centered at the origin, instead it shows you a blue speck.
Frequency: I just noticed it today.
Operating system: Windows 11
Browser: Latest version of google chrome
Additional information: This is not the first time that Asymptote breaks on AoPS... Additionally, in last nights math jam, tame the groofle, the asymptote was having trouble rendering, although I'm not sure if this is related.
![[asy]
dot((0,0));
[/asy]](//latex.artofproblemsolving.com/texer/j/jxupegwp.png)
Page URL: Any old AoPS page
Steps to reproduce:
1. Open up an asymptote window on AoPS
2. Type dot((0,0));
Expected behavior: It should show you a dot centered at the origin, instead it shows you a blue speck.
Frequency: I just noticed it today.
Operating system: Windows 11
Browser: Latest version of google chrome
Additional information: This is not the first time that Asymptote breaks on AoPS... Additionally, in last nights math jam, tame the groofle, the asymptote was having trouble rendering, although I'm not sure if this is related.
![[asy]
dot((0,0));
[/asy]](http://latex.artofproblemsolving.com/texer/j/jxupegwp.png)
![[asy]
draw(circle((0,0),5));
[/asy]](http://latex.artofproblemsolving.com/texer/r/rxuwoykf.png)
![[asy]dot((0,0));[/asy]](http://latex.artofproblemsolving.com/0/8/5/0852e3f4f3e206d3221126dc43e5f1ebc4dc95ce.png)
![[asy]dot((0,0));
[/asy]](http://latex.artofproblemsolving.com/7/b/9/7b9989ba4d53b196e4d1d5bdb2a732d393196a43.png)
![[asy]
dot((0,0));[/asy]](http://latex.artofproblemsolving.com/d/a/a/daa2c46b0c09e7a321e8469f5c0e7df2144e179b.png)
![[asy]
draw((0,0)-(1,1));
[/asy]](http://latex.artofproblemsolving.com/texer/y/ygirsehr.png)
![[asy]
draw((0,0)--(3,7),red);
dot((0,0));
dot((3,7));
label("Produced with Asymptote "+version.VERSION,point(S),2S);
[/asy]](http://latex.artofproblemsolving.com/texer/v/vzbslwxs.png)
![[asy]
draw(unitcircle);
[/asy]](http://latex.artofproblemsolving.com/texer/t/tzodflth.png)
![[asy]
dot((0,0));
[/asy]](http://latex.artofproblemsolving.com/texer/x/xsxvnwqp.png)
![[asy]
size(0.5cm);
dot((0,0));
[/asy]](http://latex.artofproblemsolving.com/a/0/4/a0432c9e2a989aeb87de8b4ec609f3f5310f95bf.png)
