[/quote]
,
be real numbers such that 
.
.
.
of the semi-circle is the midpoint of segment 
.
and 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
such that for all positive real numbers 
and 
,![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
liters of milk. Then, 
liters are removed from the container and replaced with 
and 
are fill pipes while Pipe 
is a drain pipe of a tank. Pipe 
and 
minutes to fill the remaining tank. If Pipe 
kg of these grains to the first customer. Then, it sells half of the remaining quantity and an additional 
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to 
,
,
by the recurrence![\[
a_{n+1} = a_n^2 + 1
\]](http://latex.artofproblemsolving.com/3/1/1/3114595ded8c8961d611afaab9ad19e254973001.png)
Prove that there is no natural number 
such that![\[
\prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right)
\]](http://latex.artofproblemsolving.com/a/4/f/a4f88ee7e3d4e53e369319f74cd9b12be26c9681.png)
is a perfect square.
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
;
;
;
,
;
,
is not a diameter of its circumcircle. The lines 
intersect at point 
,
and 
is acute and let 
be its orthocenter. The points 
and 
on the lines 
and 
.
,
.
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be 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?
