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and 
be relatively prime positive integers. If 
has 
positive divisors, then how many positive divisors has 
have?
has the property that, for every integer 
between 
and 
inclusive, the number 
is less than the sum of the other 
numbers. Given that 
where and are relatively prime positive integers, find 
of side length 1, let 
be a point on line 
and let 
be a point on the line through 
and the midpoint of 
. The least possible value of 
can be written as 
for relatively prime positive integers and . Find 
.
be a trapezoid with 
. Let 
and 
be the midpoints of and 
respectively, and let 
be the intersection of diagonals 
and 
. If 
, 
, 
, and 
, then 
can be expressed as for relatively prime positive integers and . Find .
denote the number of positive integers less than or equal to which are relatively prime to . Compute ![\[\sum_{i=1}^{\phi(2023)} \frac{gcd(i, \phi(2023))}{\phi(2023)}\]](http://latex.artofproblemsolving.com/0/f/8/0f8efa94170cd9e360b3daf29fe160a533baa336.png)
.
, where 
are relatively prime positive integers and is at most 
.
. A new series, obtained by squaring each term of the original series, has 
times the sum of the original series. The common ratio of the original series is where and are relatively prime integers. Find 
.
be positive reals such that 
and 
The maximum possible value of 
can be written in the form where and are relatively prime positive integers. Find the value of .
be a triangle with 
, 
, 
. Let 
be the circumcenter of 
and let be a point such that 
and 
. If lines 
and 
intersect at 
, then the length 
can be expressed in the form , where and are relatively prime positive integers. Find .
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
