AoPS Academy
of positive integers such that
be a circle inscribed inside a rhombus 
. Let 
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to . and the area of triangle 
is the same.
, find the smallest integer 
such that any graph with 
vertices and at least edges must contain at least two disjoint cycles (cycles with no common vertices).
regions of a three-set Venn Diagram with 
colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.
of them have the same color. I want to reshuffle the cupcakes such that each plate is replaced by a different color cupcake. Determine how many ways to reshuffle when:
.
.
is called beautiful if the are at least 
integers 
such that in base 
we have that and 
end in the same digit. Find the sum of the two smallest beautiful numbers.
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
and verifying 
. 
where 
are primes, show that 
can never be a perfect square.
we see immediately that this is equivalent to 
being a rational function.
for some rational function 
.![\[\frac{2}{(n+1)^2}=2r(n+1)-r(n).\]](http://latex.artofproblemsolving.com/7/5/7/757ea520dd10dbbf33afd2d31ab4a8996c771187.png)
This is true for all positive integers 
and hence must be true as an identity of rational functions identically.
and 
and that must have a pole at 
. But then must have a pole at 
, then at 
etc. so that must have infinitely many poles which is clearly a contradiction.
satisfy that 
.
and 
Prove that
Let 
Prove that
be the Fibonacci numbers, defined by 
and 
.
,
Prove that there are no polynomials 
with real coefficients such that for any positive integer 
.