[/quote]
,
such that 
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 
.
to 
.
)
.
and 
,
is not less than the area of 
,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
)
and the circle 
must hold.
the length of the line segment 
such that 
(where 
)
that satisfies 
for all 
is the function obtained by composing 
times, that is, 
)
be an acute triangle with 
,
.
past 
.
be on the perpendicular bisector of 
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
.
.
Y by Adventure10
Let 
be the point where the diagonals of a parallelogram 
meet. Consider three circles passing through : the first and the second circles are tangent to 
at points 
and 
, respectively, and the third circle passes through 
and 
. Denote the points, other than , where the third circle intersects the first and the second ones by 
and 
, respectively. Prove that 
is tangent to the first and second circles.
be the point where the diagonals of a parallelogram 
meet. Consider three circles passing through : the first and the second circles are tangent to 
at points 
and 
, respectively, and the third circle passes through 
and 
. Denote the points, other than , where the third circle intersects the first and the second ones by 
and 
, respectively. Prove that 
is tangent to the first and second circles.![[asy]
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -19.29072827281779, xmax = 31.604375765640466, ymin = -19.895374471497302, ymax = 13.031466887769763; /* image dimensions */
pen wwqqcc = rgb(0.4,0,0.8);
/* draw figures */
draw(circle((-6.1818391649118665,-6.651137429230186), 4.296681241449511), linewidth(2));
draw(circle((1.0583292112259786,-5.925050238574986), 7.742193009781517), linewidth(2));
draw((-9.729628575555873,8.336493607389526)--(-7.828586023114599,-10.619725903187096), linewidth(2));
draw((-7.828586023114599,-10.619725903187096)--(-1.9089451333227325,-13.076052655361938), linewidth(2));
draw((-1.9089451333227325,-13.076052655361938)--(-3.809987685764006,5.88016685521468), linewidth(2));
draw((-3.809987685764006,5.88016685521468)--(-9.729628575555873,8.336493607389526), linewidth(2));
draw(circle((-8.6077320752095,2.6790132746512607), 5.767645574479219), linewidth(2));
draw((-8.583853884610699,-3.088582871476122)--(-3.269862834496005,0.4943193734709459), linewidth(2) + linetype("2 2") + wwqqcc);
/* dots and labels */
dot((-5.819286854439302,-2.3697795239862067),dotstyle);
label("$M$", (-4.871197472866245,-2.3), NE * labelscalefactor);
dot((-7.828586023114599,-10.619725903187096),linewidth(4pt) + dotstyle);
dot((-8.583853884610722,-3.088582871476125),linewidth(4pt) + dotstyle);
dot((-7.828586023114599,-10.619725903187096),linewidth(4pt) + dotstyle);
label("$A$", (-8.330088038586709,-12.169021519527405), NE * labelscalefactor);
dot((-1.9089451333227325,-13.076052655361938),linewidth(4pt) + dotstyle);
label("$B$", (-1.7716721607271284,-14.684578294587372), NE * labelscalefactor);
dot((-3.809987685764006,5.88016685521468),linewidth(4pt) + dotstyle);
label("$C$", (-2.2208787277038122,5.125431309009867), NE * labelscalefactor);
dot((-9.729628575555873,8.336493607389526),linewidth(4pt) + dotstyle);
label("$D$", (-9.542945769423754,8.67416318811232), NE * labelscalefactor);
dot((-3.269862834496005,0.4943193734709459),linewidth(4pt) + dotstyle);
label("$Q$", (-3.074371204959511,0.8579689227474238), NE * labelscalefactor);
dot((-8.583853884610699,-3.088582871476122),linewidth(4pt) + dotstyle);
label("$P$", (-8.419929351982047,-2.735683613052529), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/0/6/b/06b884be1e20d17a04525e88ea16c56721a1b060.png)
are concyclic
and 
,
.
,
cyclic, we have 
,
.
and 
