holds 
.
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
.
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
Hi, everyone.
In
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.
Let the circumcircles of
, 
, 
be 
, 
, 
respectively.
Reflect in 
and then we get the circle 
Reflect in 
and then the circle 
Reflect in 
and then the circle 
Prove that , , are coaxial.
In
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.Let the circumcircles of
, 
, 
be 
, 
, 
respectively.Reflect
in 
and then we get the circle 
Reflect
in 
and then the circle 
Reflect
in 
and then the circle 
Prove that
, , are coaxial.
and Lemoine point 
.
be the reflections of 
.
are coaxial.
.
be midpoints of sides 
,
is the Anti-Steiner point of 
wrt 
.
,
,
,
are concurrent at a point (other than the midpoint of 
)
be the Euler-Poncelet point of 
,
be the midpoints of 
.
.
be the reflection of the circumcenter of 
.
being the reflection of 
across ![\[
(SM, SY) \equiv (SM, YZ) + (YZ, YS) \equiv (YZ, OG) + (OG, XY) + \frac{\pi}{2} \equiv (BC, BA) + \frac{\pi}{2} \pmod \pi
\]](http://latex.artofproblemsolving.com/a/b/4/ab41beb8d934b0ccfce78347a546b8c205ea74d3.png)
Additionally, let 
.
,![\[
(SY, SD) \equiv (AL, AC) \pmod \pi
\]](http://latex.artofproblemsolving.com/1/e/d/1ed0038d2d147de79a0d46d48eddca5c8b718d6c.png)
Therefore,![\[
(SM, SD) \equiv (BC, BA) + \frac{\pi}{2} + (AL, AC) \equiv (AC, AO) + (AL, AC) \equiv (AL, AO) \pmod \pi\]](http://latex.artofproblemsolving.com/8/4/6/846bb57c3546ec6fd67fb05319e3532f277cc8d1.png)
This implies that 
.
