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Bob writes polynomials

Source: India EGMO 2022 TST P6

November 28, 2021
algebrapolynomial

Problem Statement

Suppose P(x)P(x) is a non-constant polynomial with real coefficients, and even degree. Bob writes the polynomial P(x)P(x) on a board. At every step, if the polynomial on the board is f(x)f(x), he can replace it with
1. f(x)+cf(x)+c for a real number cc, or 2. the polynomial P(f(x))P(f(x)).
Can he always find a finite sequence of steps so the final polynomial on the board has exactly 20202020 real roots? What about 20212021?
~Sutanay Bhattacharya
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