Let R[x] be the set of all polynomials with real coefficients. Find all functions f:R[x]→R[x] satisfying the following conditions:
[*] f maps the zero polynomial to itself,
[*] for any non-zero polynomial P∈R[x], degf(P)≤1+degP, and
[*] for any two polynomials P,Q∈R[x], the polynomials P−f(Q) and Q−f(P) have the same set of real roots.
Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha PolynomialsfunctionsReal RootsalgebraINMO