AoPS Academy
+1 w
of each imon 
. During this procedure, the two copies and 
become entangled if and only if the original imons and 
are entangled, and each copy becomes entangled with its original imon ; no other entanglements occur or disappear at this moment.
members from party 
, 
members from party 
, 
members from party 
, and 
members from party 
, keeping them isolated in individual rooms in his castle, where he forces them to work.
or more people involved).
that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
at and meet at . The circumcircle of 
meets sides 
and 
again at 
and 
respectively. Let 
be the circumcentre of . Show that 
is perpendicular to 
.
and 
be chords in a circle of center with 
distinct , and with the lines and meeting at a right angle at point . Let also 
and 
be the midpoints of and 
respectively . If 
, prove that 
.
so it's contradiction. Same way for the opposite.
and 
, Again 
and 
. So 
and 
. So 
is a parallelogram. As 
, so is a rhombus. Now 
. So 
. By some angle chase we get 
