MathDB

Given three functions

P(x) = (x^2-1)^{2023}, Q(x) = (2x+1)^{14}, R(x) = \left(2x+1+\frac 2x \right)^{34}.

Initially, we pick a set

S

containing two of these functions, and we perform some operations on it. Allowed operations include:
- We can take two functions

p,q \in S

and add one of

p+q, p-q

, or

pq

S

. - We can take a function

p \in S

and add

p^k

S

for

k

is an arbitrary positive integer of our choice. - We can take a function

p \in S

and choose a real number

t

, and add to

S

one of the function

p+t, p-t, pt

.
Show that no matter how we pick

S

in the beginning, there is no way we can perform finitely many operations on

S

that would eventually yield the third function not in

S

Problem Statement