![$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$](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 
,
,
intersects line 
and 
.
is tangent to 
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
Y by Mango247, Mango247
Let 
be the altitude 
of an acute triangle. On the line are chosen different points 
and 
so that 
and point is in the interior of triangle . The circumcircle of triangle 
intersects 
and 
for second time at points 
and 
respectively. The circumcircle of the triangle 
intersects the 
and 
for the second time at points 
and 
respectively. Prove that the lines 
and 
intersect at one point.
be the altitude 
of an acute triangle. On the line are chosen different points 
and 
so that 
and point is in the interior of triangle . The circumcircle of triangle 
intersects 
and 
for second time at points 
and 
respectively. The circumcircle of the triangle 
intersects the 
and 
for the second time at points 
and 
respectively. Prove that the lines 
and 
intersect at one point.
and 
we have that 
.
.
are concyclic so 
.
.
so 
and the proof is complete.
