AoPS Academy
is polynomial with minimum degree such that 
for 
, find 
and 
such that 
and is to the left of , and replaces the pair 
by either 
or 
. Prove that she can perform only finitely many such iterations.
be non-negative real numbers such that 
. Find the maximum and minimum values of the expression![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
be reals such that 
. Prove that
Let be reals such that 
. Prove that
be a permutation of 
such that there are exactly 
ordered pairs of integers 
satisfying 
and 
. How many possible exist?
such that 
is an integer for all 
, where 
are all the positive divisors of 
.
is a prime power, which of course works as 
is always an integer. Now we prove that numbers with at least two distinct prime factors fail.
for all 
. Consider any two prime divisors 
of . Let 
, and WLOG 
. Then 
, contradiction as 
and 
are distinct primes. 
is the solution where is a prime., say and . So 
.
. So there exist the maximum power of as a factor of which is less than , say the power is 
., also 
which is a contradiction!!
such that 
.
Prove that :
Where 
and 
.
be the least of them. Divisors of 
for some 
.
does not divide