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two polynomial with real root

Source: Iran TST 2023 ; Exam 2 Problem 5

March 16, 2023
algebrapolynomialfunction

Problem Statement

Find all injective f:Z≥0→Z≥0f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 that for every natural number nn and real numbers a0,a1,...,ana_0,a_1,...,a_n (not everyone equal to 00), polynomial ∑i=0naixi\sum_{i=0}^{n}{a_i x^i} have real root if and only if ∑i=0naixf(i)\sum_{i=0}^{n}{a_i x^{f(i)}} have real root.
Proposed by Hesam Rajabzadeh
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