with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
,
,
on 
,
,
.
intersects 
at 
,
.
intersect again at 
. If 
is the tangent point of the 
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers 
with 
such that the equation 
has an integer solution 
.
such tha
Adit Aggarwal, India.
Y by itslumi, Adventure10, Mango247
In the triangle 
let 
and 
be the midpoints of the sides 
and 
respectively and 
the foot of the altitude passing through the vertex 
. Prove that the circumcircles of the triangles 
,
, and 
have a common point 
and that the line 
passes through the midpoint of the segment 
let 
and 
be the midpoints of the sides 
and 
respectively and 
the foot of the altitude passing through the vertex 
. Prove that the circumcircles of the triangles 
,
, and 
have a common point 
and that the line 
passes through the midpoint of the segment 
,
the midpoint of 
and the circle circumscribed about 
the intersection point of the line 
with the circumscribed circle about the triangle 
and 
,
,
,
,
.
(since 
)
,
,
,
are tangents to them, so we're done. 
s
and 
and 
,
(here 
means congruence since I don't know how to type congruent symbol). Now we have 
,
.
,
.
,
and let 
,
is cyclic. Indeed note that 
is a symmedian in 
invert and angle chase if you're silly like me). But then 
,
.
.
(by SAS) congruency.
.
.
