VN-TST 2023 Problem 2

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 $pq$ to $S$ .
• We can take a function $p \in S$ and add $p^k$ to $S$ 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 $S$ .