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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 
square-free partitions.
with 
intersect at 
,
and 
are the circumcenter and incenter of 
,
then 
and 
are perpendicular
be real numbers such that 
and 
,
.
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 
.
)
is a triangle. Show that 
be the common chord of two circles 
and 
.
intersects 
and 
be a point on the segment 
and 
intersect 
.
intersects 
and 
.
lie on the circumference of a circle 
.
is perpendicular to 
.
different types of stones.
has positive integer solutions.
Y by Mango247, Mango247, Mango247
A sequence 
consisting of pairwise different cells of an 
board is called a cycle if 
and cell ai shares a side with cell 
for every 
, where 
. We will say that a subset 
of the set of cells of a board is malicious if every cycle on the board contains at least one cell belonging to . Determine all real numbers 
with the following property: for every integer 
on an board there exists a malicious set containing at most 
cells.
consisting of pairwise different cells of an 
board is called a cycle if 
and cell ai shares a side with cell 
for every 
, where 
. We will say that a subset 
of the set of cells of a board is malicious if every cycle on the board contains at least one cell belonging to . Determine all real numbers 
with the following property: for every integer 
on an board there exists a malicious set containing at most 
cells.
