Show that there are infinitely many polynomials P with real coefficients such that if x, y, and z are real numbers such that $x^2+y^2+z^2+2xyz=1$, then $$P\left(x\right)^2+P\left(y\right)^2+P\left(z\right)^2+2P\left(x\right)P\left(y\right)P\left(z\right) = 1$$
Problem
Source: SAMO, 2022, R3, P6
Tags: algebra, polynomial
Wakkis1729
05.08.2022 23:35
Bump....
navid
11.12.2022 03:49
If $(x,y,z)$ satisfies the condition of the problem, i.e., $x^2+y^2+z^2+2xyz=1$ then $(P(x), P(y), P(z))$ also satisfies it. This shows that we can construct a sequence of polynomials iteratively; that is if $P(x)$ works then $P(P(x))$ works, then $P(P(P(x)))$ works, etc. So, we may only need to find a non-linear polynomial satisfying the statement of the problem. It is not so hard to find $P(x)=2x^2-1$ is the only quadratic polynomial that satisfies the statement of the problem. So, we are done.
CT17
11.06.2023 19:46
Solved with bluelinfish, CT17, CyclicISLscelesTrapezoid, v4913, rjiangbz, swimwithdolphin, psloh, tigerzhang, asdf1827, CircleInvert, SquareInvert, CT18, and asdf1434. $T_n(x)$ works for all $n$.