and the rea is 
Find the minimum of 
intersects the circumcircle of the acute-angled 
at point 
.
intersect sides 
and 
,
is the circumcenter of 
,
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 
,
Y by Adventure10, Mango247
Let 
be a rectangle, and let 
be a circular arc passing through the points 
and 
.
Let
be the circle tangent to the lines 
and 
and to the circle , and lying completely inside the rectangle .
Similiarly let
be the circle tangent to the lines 
and 
and to the circle , and lying completely inside the rectangle .
Denote by
and 
the radii of the circles and , respectively, and by 
the inradius of triangle 
.
(a) Prove that
.
(b) Prove that one of the two common internal tangents of the two circles and is parallel to the line 
and has the length 
.
be a rectangle, and let 
be a circular arc passing through the points 
and 
.Let
be the circle tangent to the lines 
and 
and to the circle , and lying completely inside the rectangle .Similiarly let
be the circle tangent to the lines 
and 
and to the circle , and lying completely inside the rectangle .Denote by
and 
the radii of the circles and , respectively, and by 
the inradius of triangle 
.(a) Prove that
.(b) Prove that one of the two common internal tangents of the two circles
and is parallel to the line 
and has the length 
.
is a triangle and 
is a circle passing through 
is a circle,which is tangent to 
at points 
and 
is incenter of 
lie on the same circle and 
lie on the same circle.(it is true in two cases on picture).
