be a positive integer. We start with 
and 
be the incircle and circumcircle of the acute triangle 
so that all of its sides are tangent to 
,
are both on 
.
and 
,
and 
,
and 
intersects on 
be integers greater than 
,
be positive integers not greater than 
.
not greater than ![\[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \]](http://latex.artofproblemsolving.com/7/4/1/7414caaacc9fd761961279f8a37ab6cd6af6b49a.png)
where 
denotes the greatest common divisor of 
.
be a fixed integer. Let 
and for all 
let 
.
for which sequence 
will be bounded?
denotes Euler’s Phi Function, meaning that 
gives the number of integers within the set 
that are relative primes to 
bounded if there exist a constant 
such that 
for all positive integers 
Y by Mango247
Let 
be an acute, non-isosceles triangle with the circumcircle 
. Denote 
as the midpoints of 
respectively. Two circles 
and 
intersect at 
differs from 
. Suppose that the ray 
intersects at 
. The line 
meets at the second point 
and the line 
meets at the second point 
.
a) Prove that
collinear and 
perpendicular to 
.
b) Prove that is the midpoint of
be an acute, non-isosceles triangle with the circumcircle 
. Denote 
as the midpoints of 
respectively. Two circles 
and 
intersect at 
differs from 
. Suppose that the ray 
intersects at 
. The line 
meets at the second point 
and the line 
meets at the second point 
.a) Prove that
collinear and 
perpendicular to 
.b) Prove that
is the midpoint of 
and 
and 
,
,
cyclic. If 
then 
with this a) and b) are almost immediate, for a) see that 
is midpoint of 
so 
but 
so 
lie in a circle with diameter 
.
is a parallelogram.
?By angle chase, ended up being unable to find an angle that equals 
