or 
.
and 
,
and 
operated by two different companies, then there must exist flight route 
operated by the third company different from 
,
and 
denote the number of flight routes operated by companies 
.
such that![\[f(x^{n+1}+y^{n+1})=x^nf(x)+y^nf(y),\forall x,y>1.\]](http://latex.artofproblemsolving.com/a/4/7/a47d53806f2148cc3339335fb990b33641085896.png)
be a fixed integer. Find the least constant ![\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]](http://latex.artofproblemsolving.com/d/d/e/dde2d1dd0c412d8ba640f1d4c044e2fa2dcccc82.png)
(the sum on the left consists of 
summands). For this constant 
,
be the least prime that does not divide 
,
to be the product of all primes less than 
For 
,
.
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
.
.
Y by
Given a polynomial 
and a positive integer 
, we denote the -fold composition of by 
. A polynomial with real coefficients is called perfect if for each integer 
there is a positive integer so that 
is an integer. Is it true that for each perfect polynomial , there exists a positive 
so that for each integer there is 
for which is an integer?
and a positive integer 
, we denote the -fold composition of by 
. A polynomial with real coefficients is called perfect if for each integer 
there is a positive integer so that 
is an integer. Is it true that for each perfect polynomial , there exists a positive 
so that for each integer there is 
for which is an integer?This post has been edited 1 time. Last edited by Phorphyrion, Apr 5, 2023, 1:48 PM
and 
is a fixed point.
,
such that the ![\[
P(x) = \frac{4x^3 + 6x^2 + 10x + 3}{2}
\]](http://latex.artofproblemsolving.com/e/9/5/e9576fb4152ac11bea39f043b1f050b0fd81c159.png)
evaluated at ![\[
P\left(\frac{n-1}{2}\right) = \frac{n(n^2 + 7)}{4} - \frac{1}{2}.
\]](http://latex.artofproblemsolving.com/6/f/a/6fa565afb529efed4a90a79c5ddc2977bce75e71.png)
.
must be a half-integer, i.e., 
for some odd integer 
.![\[
2P(n) + 1,\quad 2P(P(n)) + 1,\quad 2P(P(P(n))) + 1, \ldots
\]](http://latex.artofproblemsolving.com/9/b/3/9b3324db81c2cce4cf53ce05ea7fb76515fe42bb.png)
![\[
m \mapsto (m^3 + 7m)/2.
\]](http://latex.artofproblemsolving.com/d/e/e/dee7010b7d067c28abf03240e9b31d2825660684.png)
Hence, the sequence follows the recurrence:![\[
a_{i+1} = (a_i^3 + 7a_i)/2.
\]](http://latex.artofproblemsolving.com/3/f/7/3f76a3ede66c73f59e183a0e70aaa41cc5ba1d3f.png)
![\[
v_2((m^3 + 7m)/2) = v_2(m) - 1,
\]](http://latex.artofproblemsolving.com/3/3/6/3361ac1d7359aa70f7fd71fd34e5a4db106c543d.png)
where 
denotes the 2-adic valuation. That is, each iteration reduces the 2-adic valuation by 1. Therefore, after exactly 
steps, the resulting term in the sequence will be odd. Since the input to 
at that point is an integer, and 
for odd inputs, the composition 
.
.![\[
2P\left(\frac{n-1}{2}\right) + 1 = n(n^2 + 7),
\]](http://latex.artofproblemsolving.com/b/b/9/bb9167c42684c20e5c4941bddc513e1de22a69fc.png)
we note that 
for odd 
is arbitrarily large. It is known that 
is a quadratic residue modulo any power of 2, so there exist such 
