[/quote]
,
,
and 
lie on the side 
,
.
is the midpoint of 
.
.
.
Let 
.
be a positive integer. We say that a positive integer 
is 
is divisible by 
for all positive integers 
.
is not 
is prime.
,
.
is the inverse of function 
and thus, they might intersect that 
axis (if they do, we are done). Thus: 
is the only solution.
and 
be the circumcenter and the orthocenter. The point 
is the reflection of 
with respect to 
.
lie on 
respectively such that 
.
be the intersection point of 
.
.
.
a grid of 9*9, we choose a little square in 
be the incenter of triangle 
.
,
,
be the orthocenters of triangles 
,
,
,
,
,
,
.
.
be 
.
,
,
,
,
,
similarly. Prove that 
,
,
are concurrent.
be real numbers such that 
. Find the value of 
and find at least one triplet 
that satisfy those conditions.
be 
,
.
?
,
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
such that 
.
Y by
Let 
denote the number of positive integers less than
that are relatively prime to 
. Let 
denote the set of positive integers such that when 
is expressed as a simplified fraction, the denominator is a power of 
. Find the smallest prime number 
that satisfies the following:
is not divisible by any square of a positive integer greater than 
,
No element in is divisible by .
denote the number of positive integers less than
that are relatively prime to 
. Let 
denote the set of positive integers such that when 
is expressed as a simplified fraction, the denominator is a power of 
. Find the smallest prime number 
that satisfies the following:
is not divisible by any square of a positive integer greater than 
, No element in is divisible by .
and the denominator when simplified must be a power of 2 for 
cannot be divisible by 
.
such that the simplified denominator of 
is not be divisible by any prime that is not 
.
and 
in the 
.
to not be divisible by any prime that is not 
The denominator is divisible by 
,
so it does indeed satisfy the conditions.
.
fail the first condition, leaving us with 
.
,
and thus the simplified denominator is a power of 
,
and the simplified denominator is a power of 
