AoPS Academy
be non-negative real numbers such that 
. Prove that 
be a positive integer. Find the number of sequences 
of integers in the range ![$[0,n]$](http://latex.artofproblemsolving.com/1/1/e/11ef480c3077b93f3ac006b998c520ece6100116.png)
such that for all integers 
and all nonnegative integers 
, there exists an integer 
such that 
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.
be a triangle with 
. Let 
be a point on the line 
beyond 
such that 
. Let 
be the mid-point of 
and let 
be the point on the side such that 
. Prove that 
be a point on the line , above such that 
=
is the midpoint of side 
,
is parallel to 
(By midpoint theorem's converse)
as 
and angle chasing
=
=
is isosceles
which gives 
>
allows us to construct the desired line-segment and finish the problem
and 
Prove that
is isosceles.
to 
beyond 
.
.
.
.
.
(exterior angle).
,
.
.