be the Fermat point of a 
.
,
,
are concurrent and the point of concurrence is 
,
(
well). i want to study so if you guys have some easy or normal problems please send me
?
chess board, except for the 
central square, there is a checker. In one move, a checker can jump over adjacent checker (diagonally), and this adjacent checker (that was jumped over) is removed. Is it possible to come up with a sequence of moves, so that only one checker is left on the board?
Y by
In the quadrilateral 
the angles 
and 
are straight. The lines 
and 
intersect at 
and the lines 
and 
intersect at 
The line passing through parallel to 
D intersects the circumscribed circle 
of 
at 
and the segment 
intersects at 
Prove that the line 
divides the segment 
in half.
the angles 
and 
are straight. The lines 
and 
intersect at 
and the lines 
and 
intersect at 
The line passing through parallel to 
D intersects the circumscribed circle 
of 
at 
and the segment 
intersects at 
Prove that the line 
divides the segment 
in half.
.
is the reflection of 
over 
,
is cyclic, since this would imply 
.
,
is orthocentre of 
.
-
.
These two conditions force that 
.
.
denote the feet of altitudes from 
respectively. The line through 
again at some point 
are collinear.
denote the orthocentre and 
the midpoint of 
.![\[\angle SPE = \angle SFE = \angle SBC\]](http://latex.artofproblemsolving.com/3/4/5/3457c99495ab17aa121e3b7110299fbf86ba21c0.png)
Since 
,
are collinear, and we are done. 
since 
.
thus, 
q
