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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 
that satisfy ![$f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$](http://latex.artofproblemsolving.com/4/6/c/46ca3f421006e12efc7c1501609e187dd8f0b760.png)
for all real 
.![$\sqrt[3]{x}$](http://latex.artofproblemsolving.com/d/d/2/dd20cd4b23d9242a4349d63b1de7514bc295f39c.png)
is defined all over 
.)
the substitutions 
and ![$x =\sqrt[3]{1-t^3}$](http://latex.artofproblemsolving.com/7/7/1/7715cc1eefe99ceb73ae0cee59d98f83288d3d3d.png)
give the following system linear in 
and ![$f(\sqrt[3]{1-t^3})$](http://latex.artofproblemsolving.com/8/b/9/8b90082c3f28727ac67cac85d72d040575910d78.png)
:![$$f(t) + 2f(\sqrt[3]{1-t^3}) = t^3,$$](http://latex.artofproblemsolving.com/b/5/e/b5ec5f3fb48ceb042111a23547e1080f1a9ab034.png)
![$$f(\sqrt[3]{1-t^3})+2f(t) = 1-t^3,$$](http://latex.artofproblemsolving.com/8/0/6/806e290b77c5e7f3b0dcc6643ddd65affefdb886.png)
which gives the unique solution 
,