and 
are internally tangent at point 
.
of 
,
is tangent to 
and 
and 
respectively. Rays 
and 
.
is the midpoint of major arc 
.
and two fixed points 
on that circle, let 
be a moving point on 
is acute and scalene. Let 
be the midpoint of 
and let 
be the three heights of 
,
such that 
.
be the intersection of 
and 
,
be the second intersection of 
and 
.
always goes through a fixed point.
intersects 
.
of 
,
such that 
.
be the center of 
.
always goes through a fixed point.
where 
Y by
Let 
be a finite set of cells in a square grid. On each cell of we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of still contains one grasshopper.
be a finite set of cells in a square grid. On each cell of we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of still contains one grasshopper.
- Prove that, for every set , the number of Asturian arrangements is a perfect square.
- Compute the number of Asturian arrangements if is the following set:
Attachments:
and color 
with 
.
such that 
for all cells 
.
and 
depending on color, and realise that it is sufficient to find bijections 
and 
such that 
for all 
and 
for all 
.
are essentially the same thing (the inverse of each 
is a valid 
and similarily for the other way round). As a result, we obtain that if there are 
bijections 
,
