![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](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 
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
The circle inscribed in triangle 
has center 
and touches sides 
and 
at points 
and 
respectively. The perpendicular bisector of segment 
intersects side at point 
. Through point , a line perpendicular to 
is drawn, it intersects side at point 
. Prove that lines and 
are perpendicular.
has center 
and touches sides 
and 
at points 
and 
respectively. The perpendicular bisector of segment 
intersects side at point 
. Through point , a line perpendicular to 
is drawn, it intersects side at point 
. Prove that lines and 
are perpendicular.This post has been edited 1 time. Last edited by parmenides51, May 29, 2022, 11:24 PM
