AoPS Academy
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 
.
for which 
is the square of an integer
. How many complex numbers 
are there such that 
and 
is an integer. and be points on a plane such that 
, where is a positive integer. Let 
be the set of all points 
such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
and 
are defined in the following manner:
then find the value of 
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
be the circumcenters of the 2 triangles respectively. We have 
.
and 
Prove that
then find the value of 
be the intersection point of the medians 
and 
of a right triangle 
)
and 
are tangent. Find the angle 
