[/quote]
,
such that every collection of 
.
,
on line 
such that 
are in that order and 
and 
.
on line 
,
on line 
.
and 
are equidistant from the circumcenter of 
be six positive real numbers. Prove that![\[a^2+b^2+c^2+\frac{4(ax+by+cz)\sqrt{ab+bc+ca}}{x+y+z} \geqslant 2(ab+bc+ca).\]](http://latex.artofproblemsolving.com/f/8/0/f808f7857c2150960f2a430837286487666797d8.png)
be a cyclic quadrilateral with circumcentre 
and circumcircle 
.
be the intersection of tangents at 
and 
to 
be the circumcircle of triangle 
and let 
,
denote the second intersections of 
with 
and 
and 
be the circumcircle of triangle 
.
,
is also tangent to 
.
be the midpoint of 
be the intersection of the tangents from 
.
intersects 
.
and 
meet at 
,
and 
meet at 
.
be a positive integer. Alice and Bob play the following game, with Alice going first after which they alternate turns. They determine the numbers 
in the following way.
to be an integer such that:
for 
.
successfully.
for which Alice can ensure that she wins regardless of Bob's strategy.
in the circumcircle of 
the point where the 
-
meets 
and intersects 
or 
is tangent to 
and 
are tangent to the circumcircle of 
be the reflection of 
be the reflection of 
,
is perpendicular to 
denote the set of all positive integers. Find all real numbers 
for which there exists a function 
satisfying:
,
is an integer if and only if 
;
,
.
be a positive integer larger than 
,
be integers. It is known that the equation 
has 
and 
are relatively prime.
Y by
asking for a friend who is designing their own video classroom....
think back to when you had online classes with video (like on zoom and stuff):
do you like the chat feature that they have?
what did you use the chat for?
what would you do instead if there wasn't a chat?
what other thoughts do you have about having chat in a video classroom?
think back to when you had online classes with video (like on zoom and stuff):
do you like the chat feature that they have?
what did you use the chat for?
what would you do instead if there wasn't a chat?
what other thoughts do you have about having chat in a video classroom?
