AoPS Academy
, denote the sum of digits of in its decimal representation by 
. Find all polynomials 
with integer coefficients such that for any positive integer 
, the integer 
is positive and 
be a triangle with incentre 
and circumcircle 
. Let 
be the midpoints of the arcs 
of not containing 
respectively. Let 
be the point of diametrically opposite to 
. Show that 
and the midpoint 
of 
lie on a line.
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
is a trapezoid, 
. The sides 
, 
, 
are tangent to 
and touches at 
. The sides , 
, are tangent to 
, and touches at 
. Prove that the lines 
, 
, 
meet at the same point.![[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(10pt));
pair A=origin,B=(1,-7),C=(30,-15),D=(26,6);
pair bisA=bisectorpoint(B,A,D),bisB=bisectorpoint(A,B,C),bisC=bisectorpoint(B,C,D),bisD=bisectorpoint(C,D,A);
path bA=A--(bisA+100*(bisA-A)),bB=B--(bisB+100*(bisB-B)),bC=C--(bisC+100*(bisC-C)),bD=D--(bisD+100*(bisD-D));
pair O1=intersectionpoint(bA,bB),O2=intersectionpoint(bC,bD);
dot(O1^^O2,linewidth(2));
pair h1=foot(O1,A,B),h2=foot(O2,C,D);
real r1=abs(O1-h1),r2=abs(O2-h2);
draw(circle(O1,r1)^^circle(O2,r2));
draw(A--B--C--D--cycle);
draw(A--C^^B--D^^h1--h2);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,dir(350));
label("$D$",D,dir(350));
label("$P$",h1,dir(200));
label("$Q$",h2,dir(350));
label("$O_1$",O1,dir(150));
label("$O_2$",O2,dir(300));
[/asy]](http://latex.artofproblemsolving.com/2/1/3/21358d59310e2109169924c2c5f7900ef449be9b.png)
and 
Let 
be the incircle of 
touching at 
Clearly 
and 
are homothethic with incircles 
but since 
are symmetric about the midpoint of 
( is the M-excircle of MAB), it follows that 
and 
are homothetic with corresponding cevians 
is a trapezoid, . The sides , , are tangent to and touches at . The sides , , are tangent to , and touches at . Prove that the lines , , meet at the same point.
is the intersection of and .
, , meet at the same point .
and at Y and T. Let KB meet and at X and R.
.
and 
are tangent from 
, 
.
and 
Since 
and 
we see 
where 
and 
denote the semiperimeter of 
and 
respectively. Since 
we know 
