![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](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 
and 
.
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
An open disc (that is to say without its edge) of radius 
is divided into open regions by two lines.
(a) If the two lines are parallel, is there still an open disc of radius
which is entirely contained in one of these regions?
(b) If the two lines are perpendicular, is there still an open disc of radius which is entirely contained in one of these regions?
(c) Is there always an open disc of radius which is entirely contained in one of these regions, regardless of the position of the two lines?
is divided into open regions by two lines.(a) If the two lines are parallel, is there still an open disc of radius
which is entirely contained in one of these regions?(b) If the two lines are perpendicular, is there still an open disc of radius
which is entirely contained in one of these regions?(c) Is there always an open disc of radius
which is entirely contained in one of these regions, regardless of the position of the two lines?This post has been edited 1 time. Last edited by parmenides51, Aug 10, 2020, 11:11 AM
