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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.
be the sequence of Fibonacci numbers: 
, 
and 
, for every 
.
, divides 
.
such that 
![$\sqrt[3]{1342\sqrt{167}+2005}$](http://latex.artofproblemsolving.com/a/4/0/a40560c583ff14eadbd5e1f4c4207d98119234dc.png)
in the form where it contains only addition, subtraction, multiplication, division and square roots.
