![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](http://latex.artofproblemsolving.com/a/a/6/aa671ae30c20a584500d6362790124ad6cbb67be.png)
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 
,
,
be the excircle opposite to 
.
and 
be the points where 
,
,
intersects line 
at 
be the midpoint of 
.
is tangent to 
be reals such that 
. Prove that
Let 
. Prove that
.
are the midpoints of 
respectively. 
,
.
be the incenter and tangent of angles 
respectively. Prove that 
have a common point
and 
.
Y by
Let 
, 
be two parallel lines and 
a line perpendicular to , and to at 
and 
, respectively. Points 
and 
lie on such that 
(
). Let 
be the diameter of the circle inscribed in the triangle 
. Point 
lies on in the same half plane as wrt the line such that: 
Let 
be the intersection point of 
and 
. Find the locus of points as point moves along with the exception of .
, 
be two parallel lines and 
a line perpendicular to , and to at 
and 
, respectively. Points 
and 
lie on such that 
(
). Let 
be the diameter of the circle inscribed in the triangle 
. Point 
lies on in the same half plane as wrt the line such that: 
Let 
be the intersection point of 
and 
. Find the locus of points as point moves along with the exception of .This post has been edited 3 times. Last edited by parmenides51, Oct 23, 2021, 4:54 AM
Reason: typo: O->Q
Reason: typo: O->Q
and 
.
,
with 
.
.
.
.
.
,
,
.
