AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be an acute triangle with orthocenter 
. Let 
be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that 
bisects 
. Suppose that the line intersects the circumcircle of the triangle 
at 
and 
. Prove that 
.
of the form 
with 
, consider the polynomial 
Determine all ordered pairs of polynomials 
with integer coefficients such that 
.
be a polynomial of degree 
with only real zeros and real coefficients.
we have 
. When does equality occur?
and 
and move both of them to the midpoint of and . tokens is collapsible if it is possible to end up with all tokens at the same point after a finite number of moves. Prove that every arrangement of tokens is collapsible if and only if is a power of 
.
be real numbers.Prove that
be an 
square matrix with eigenvalues 
(each 
having algebraic multiplicity 
, so that 
). Let 
be a polynomial. It is known that if 
is an eigenvalue of then 
is an eigenvalue of 
. of the form 
(counting multiplicities)?
vs. equation 
)
where 
denotes the Fibonacci sequence.**
satisfying ![\[|z -a | + |z + a| = 2|c|\]](http://latex.artofproblemsolving.com/4/b/d/4bdafe46f8d3ed1c8945a1553fc3bc0d23fe11e2.png)
if and only if 
If this condition is fulfilled, what are the smallest and largest values of 
and radius 
and 
.
(with multiplicity 3),
(with multiplicity 2),
(with multiplicity 1),
(with multiplicity 3), 
(with multiplicity 2), and 
(with multiplicity 1)?
