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
to 
is 
respectively and they concur at 
And let 
meet the line at 
. We are going to show that 
.
; 
and 
are cyclic. So if we write powers of wrt these circle we get![\[ AP\cdot PA'=BP\cdot PB'=CP\cdot PC'=DP\cdot PD'=PR^2 \]](http://latex.artofproblemsolving.com/6/4/d/64dca97f5c37538ec0f352a78995fe24c93e097c.png)
Let 
be the 
-radius circle with center . Inversion wrt swaps with respectively. So the line maps to 
.
is cyclic.
and 
are orthologic triangles with ortgology center 
![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](http://latex.artofproblemsolving.com/a/a/6/aa671ae30c20a584500d6362790124ad6cbb67be.png)
and 
.