AoPS Academy
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
, let 
be the excircle opposite to 
. Let 
and 
be the points where is tangent to 
, and 
, respectively. The circle 
intersects line 
at 
and 
. Let 
be the midpoint of 
. Prove that the circle 
is tangent to .
be reals such that 
. Prove that
Let be reals such that 
. Prove that
. 
are the midpoints of 
respectively. is the midpoint of arc 
, is symmetric to through 
. Let 
be the incenter and tangent of angles of triangle 
respectively. Prove that 
have a common point
be a point inside 
. Let the perpendicular bisectors of 
be 
. Let 
, 
, 
. If 
lie on a circle, prove that 
are collinear.
, 
and 
are on a circle, we have 
(as 
is the perpendicular bisector of 
)
(as 
is the perpendicular bisector of 
)
be the incenter of triangle . It is obvious that 
lies on the circumcircle of triangle 
. lies on the circumcircle of triangle 
and circumcircle of triangle 
. and coincide.
![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](http://latex.artofproblemsolving.com/a/a/6/aa671ae30c20a584500d6362790124ad6cbb67be.png)
and 
.
,
,
,
)
,
.
are collinear (and so are 
,
)![[asy]
pair D = dir(105); pair E = dir(195); pair F = dir(345);
pair G = foot(D, E, F); pair H = foot(E, D, F); pair I = foot(F, D, E);
pair P = extension(D, G, E, H);
pair C = 2G - P; pair B = 2H - P; pair A = 2I - P;
draw(circumcircle(D, E, F), lightblue); draw(D--E--F--cycle, magenta); draw(E--P, dotted+deepmagenta); draw(D--P, dotted+deepmagenta); draw(F--P, dotted+deepmagenta); draw(P--A, deepmagenta); draw(P--B, deepmagenta); draw(P--C, deepmagenta); draw(A--B--C--cycle, darkmagenta);
dot("D", D, dir(D)); dot("E", E, dir(E)); dot("F", F, dir(F)); dot("P", P, dir(P)); dot("C", C, dir(C)); dot("B", B, dir(B)); dot("A", A, dir(A));
[/asy]](http://latex.artofproblemsolving.com/f/a/a/faa4b32a2fda9e028447655e7a492f5201ef833e.png)
is the circumcenter of 
it must then also be the midpoint of minor arc 
applying the same logic to 
and 
