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be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
such that 
+1 w
sastifying 
the inscribed circle touches sides 
, 
and 
, at points 
, 
and 
respectively. Denote the orthocenters of the triangles 
and 
as 
and 
. Prove that the quadrilateral 
is cyclic.
the inscribed circle touches sides , and , at points , and respectively. Denote the orthocenters of the triangles and as and . Prove that the quadrilateral is cyclic.
be the incenter of the triangle. It is easy to get that 
. Thus ![\[\frac{IH_1}{IH_2}=\frac{\sin \frac{A}{2}}{\sin \frac{C}{2}}=\frac{IC}{IA}.\]](http://latex.artofproblemsolving.com/7/9/7/797e53926836482c2531e2944eb434268706567c.png)
Thus is cyclic.
and 
. Also let 
be the inradius of 
. Note that is the reflection of about 
, and is the reflection of about 
. This gives 
Similarly, we have 
, giving that is cyclic.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
to find that![\[\angle H_1AC+\angle CH_2H_1=\angle C_1AH_1+180^\circ-\angle H_1H_2I=180^\circ+\angle C_1AI-\angle C_1B_1I=180^\circ\]](http://latex.artofproblemsolving.com/e/3/2/e32e96e9792a2c81a301bf33b846367ab0b682f9.png)