[/quote]
,
be real numbers satisfying 
Prove that 
and determine all cases of equality.
are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. 
numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values.
ever be a square for positive integers 
,
,
?
is tangent to sides 
and 
at 
and 
respectively, and 
is the circumcenter of 
.
.
be an acute-angled and scalene triangle, and 
be a point on the side 
.
and 
are taken on 
such that 
and 
.
and 
are taken on 
and 
.
intersects 
for the second time at 
,
intersects 
for the second time at 
.
,
,
be a trapezoid with 
.
such that 
and 
.
be a point on 
such that 
.
to 
bisects the segment 
.
be a circumscribed quadrilateral with circumcircle 
such that 
,
.
is taken on 
.
is the reflection of 
,
is tangent to the line 
.
the foot of the 
be an arbitrary point on the arc 
from 
is such that 
.
is the reflection of 
with respect to 
intersects 
,
are concyclic.
.
passes through a fixed point.
![\[\frac{1}{4a^2+bc}+\frac{1}{4b^2+ca}+\frac{1}{4c^2+ab} \geqslant \frac{9}{a^2+b^2+c^2+4(ab+bc+ca)}.\]](http://latex.artofproblemsolving.com/e/7/7/e77a05f7638785fcf3bc7a710b25dc1fd6470b26.png)
such that ![\[f(x^2)+2xf(y)=yf(x)+xf(x+y).\]](http://latex.artofproblemsolving.com/a/f/8/af86cbfd2f518bd0cb42a1f17bc39c9fd6272208.png)
Y by
Summary of the problem: The Asymptote code at https://artofproblemsolving.com/texer/igdncqbp does not load and instead shows the following error message: "Error making remote request. Timeout error. Sorry, rendering operation timed out. Please try again in a few minutes."
Page URL: https://artofproblemsolving.com/texer/igdncqbp
Steps to reproduce:
1. Click the link above.
2. Click "Render as Image". Clicking "Render with bbCode" will show "Unrecognized services response." after 
seconds instead.
3. Wait for seconds.
4. Observe that the aforementioned error message appears.
Expected behavior: The Asymptote code renders correctly.
Frequency:
throughout two days.
Operating system(s): Windows 11, version 22H2, build 22621.1848.
Browser(s), including version: Microsoft Edge, 114.0.1823.67, 64-bit.
Additional information: Even though the end of the error message seems to suggest that waiting for a few minutes and then rendering will work, this has not worked on multiple occasions.
Page URL: https://artofproblemsolving.com/texer/igdncqbp
Steps to reproduce:
1. Click the link above.2. Click "Render as Image". Clicking "Render with bbCode" will show "Unrecognized services response." after 
seconds instead.3. Wait for seconds.4. Observe that the aforementioned error message appears.Expected behavior: The Asymptote code renders correctly.
Frequency:
throughout two days.Operating system(s): Windows 11, version 22H2, build 22621.1848.
Browser(s), including version: Microsoft Edge, 114.0.1823.67, 64-bit.
Additional information: Even though the end of the error message seems to suggest that waiting for a few minutes and then rendering will work, this has not worked on multiple occasions.
