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with 
intersect at 
, and 
and 
are the circumcenter and incenter of 
, respectively. Prove that if 
then 
and 
are perpendicular
be real numbers such that 
and 
, for all 
. Prove that 
for all 
for which the number 
is a perfect square.
such that the centroid of the triangle lies on the circle inscribed in the triangle. Prove that 
we can find positive integers 
such that 
. Show that in any such equation there must be two of the 
numbers with a common divisor (
).
is a triangle. Show that 
be the common chord of two circles 
and 
. A line through 
intersects at 
and at 
. Let 
be a point on the segment , different from 
and . The line 
intersect at and 
. The line 
intersects at 
and 
.
lie on the circumference of a circle 
. is , prove that 
is perpendicular to 
.
inhabitants of an island, where is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends.
different types of stones. types.
for which the equation 
has positive integer solutions.
, consider the numbers 
is even. for which 
and 
are primes.
teams (every team plays with every other team for a win or loss), there exist 
teams such that the first seven teams have each defeated the remaining teams.
teams (every team plays with every other team for a win or loss), there exist teams such that the first seven teams have each defeated the remaining teams.
