and 
. Prove that
Where 
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 
,
is a connected graph with 
edges. Prove that it is possible to label the edges 
in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
belongs to at most one edge. The graph 
is connected if for each pair of distinct vertices 
there is some sequence of vertices 
such that each pair 
is joined by an edge of 
.
and 
. Prove that
Where 
probability of coming up heads and 
probability of coming up tails. The expected number of flips necessary to first see the sequence 
in that consecutive order can be written as 
for relatively prime positive integers 
,
be four distinct points in the plane.
and 
are distinct and parallel, then
lies on the segment 
,
plus the oriented angle 
e
.
equals 0°, then
a
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
be reals such that 
. Prove that
specifies the greatest common divisor of 
and 
.
and 
Prove that
Let 
Prove that
Y by Adventure10, Mango247
Let 
be an acute triangle with the circumcircle 
and incenter 
. The perpendicular through in 
intersects segment ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
in 
and in 
. In particular and 
are on different sides of 
. The parallel line through to 
intersects 
in 
.
Prove: If
and are perpendicular to each other, then intersects segment ![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
in its midpoint 
.
(Notation:![$[\cdot]$](//latex.artofproblemsolving.com/5/7/9/579a0ec9e030cc17f3b2c9cd1510c9493b1813b2.png)
denotes the line segment.)
be an acute triangle with the circumcircle 
and incenter 
. The perpendicular through in 
intersects segment ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
in 
and in 
. In particular and 
are on different sides of 
. The parallel line through to 
intersects 
in 
.Prove: If
and are perpendicular to each other, then intersects segment ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
in its midpoint 
.(Notation:
![$[\cdot]$](http://latex.artofproblemsolving.com/5/7/9/579a0ec9e030cc17f3b2c9cd1510c9493b1813b2.png)
denotes the line segment.)
and 
.
are concyclic ,
is the midpoint of arc 
in 
is the perpendicular bisector of 
,
.
is the midpoint of arc 
(contain 
)
,
,
are collinear 
passes through the midpoint 
of 
.
