be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
such that 
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 
.
sastifying 
Y by
The point 
, inside the triangle ![$[ABC]$](//latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
, lies on the perpendicular bisector of ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. 
and 
points, exterior to the triangle, they are such that ![$ [BPA], [BQC]$](//latex.artofproblemsolving.com/2/c/8/2c8b6a2fbd4f51c48f808f700b1af3fae0936cf1.png)
and ![$[CRA]$](//latex.artofproblemsolving.com/8/c/9/8c9ecb973ec09cadd4405425df0417261916e78f.png)
are similar triangles. Shows that ![$[PQCR]$](//latex.artofproblemsolving.com/2/2/f/22fdb067ee7ef2dca7761127fcffd8ed6fba5341.png)
is a parallelogram.
, inside the triangle ![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
, lies on the perpendicular bisector of ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. 
and 
points, exterior to the triangle, they are such that ![$ [BPA], [BQC]$](http://latex.artofproblemsolving.com/2/c/8/2c8b6a2fbd4f51c48f808f700b1af3fae0936cf1.png)
and ![$[CRA]$](http://latex.artofproblemsolving.com/8/c/9/8c9ecb973ec09cadd4405425df0417261916e78f.png)
are similar triangles. Shows that ![$[PQCR]$](http://latex.artofproblemsolving.com/2/2/f/22fdb067ee7ef2dca7761127fcffd8ed6fba5341.png)
is a parallelogram.This post has been edited 2 times. Last edited by parmenides51, Oct 10, 2020, 6:59 AM
with angle chasing and similarly 
is a parallelogram.
.
,
.
and 
,
