and 
where each summation runs over 
variables 
and 
Then find the value of 
Prove that
[/quote]
such that 
.
Where 
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
and 
Prove that 
be real numbers such that 
.
Prove that 
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
:
be the altitude and 
be the intersection point of the altitudes of triangle 
.
is symmetric to 
.
intersects 
and 
.
.
If the value of 
can be written as 
where 
and 
where 
and 
where 
.
lie on the same circle. If 
and 
,
,
or 11. However, I didn't know how to find the radius. Can anyone give me a hint?
then find the value of 
be a sequence of reals such that 
and 
for all positive integers 
approaches the real number 
.
Y by centslordm
The function 
denotes the number of positive integers less than or equal to 
that are relatively prime to 
Determine the number of positive integers 
relatively prime to 
such that there exists some integer satisfying
William Dai
denotes the number of positive integers less than or equal to 
that are relatively prime to 
Determine the number of positive integers 
relatively prime to 
such that there exists some integer satisfying
William DaiThis post has been edited 3 times. Last edited by innumerateguy, Jul 24, 2021, 11:30 PM
I believe
.
.
.
work so 
.
.
,
where the LHS's three factors could be multiplied in or not. Therefore, the answer is 
.
to get the denominator of 
.
do not divide 
,
and 
to clear the denominator, so that is why.
,
makes 
.
,
.
to be a distinct positive integer, 
must be a distinct value that divides 
since 
is an integer, and 
is positive. Now, associating any of these numbers results in a distinct output, so the total amount of sets for these 
numbers are 
