,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
Prove that
pairwise distinct positive integer numbers 
,![\[
\frac{a_i^2 + a_j^2}{a_i + a_j}
\]](http://latex.artofproblemsolving.com/3/5/3/3537ddec6d793d3831ba82a1e13b98d43b1c29bc.png)
students in the IMO team selection camp.
team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these 
,
,
.
be an arbitrary point inside 
such that 
.
intersects 
(other than 
)
(other than 
.
intersects rays 
and 
at points 
and 
,
.
be a cyclic quadrilateral with 
and 
.
and 
are selected on segment 
.
and 
are selected on segment 
such that 
.
is a cyclic quadrilateral.
be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
.
.
Y by
A polynomial 
with degree 
is reflexive if there is an integer 
such that 
for every 
, where 
for 
. Let 
be an integer and 
be a polynomial with integer coefficients. Prove that there exist reflexive polynomials 
, 
with integer coefficients such that
![\[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]](//latex.artofproblemsolving.com/6/6/7/66729b54a8b87c6ddbfdf7e7e5932dbb8b262439.png)
with degree 
is reflexive if there is an integer 
such that 
for every 
, where 
for 
. Let 
be an integer and 
be a polynomial with integer coefficients. Prove that there exist reflexive polynomials 
, 
with integer coefficients such that![\[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]](http://latex.artofproblemsolving.com/6/6/7/66729b54a8b87c6ddbfdf7e7e5932dbb8b262439.png)
such that 
is a polynomial. Now we state that taking:
works. First we see that both are polynomials because the numerators have roots at 
.
Finally we compute:
