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positive integers into 
pairs is called square-free if the product of numbers in each pair is not a perfect square.Prove that if for distinct positive integers, there exists one square-free partition, then there exists at least 
square-free partitions.
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.
such that for all pairs 
of real numbers with 
and 
, 
Also determine for which pairs equality holds.
be real numbers such that 
Equality holds when 
or 
Equality holds when 
Equality holds when 
or 
