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Polynomial and absolute value

Source: 2018 Greece National Olympiad Problem 3

March 3, 2018
algebrapolynomialabsolute value

Problem Statement

Let n,mn,m be positive integers such that n<mn<m and a1,a2,...,ama_1, a_2, ..., a_m be different real numbers. (a) Find all polynomials PP with real coefficients and degree at most nn such that: ∣P(ai)−P(aj)∣=∣ai−aj∣|P(a_i)-P(a_j)|=|a_i-a_j| for all i,j={1,2,...,m}i,j=\{1, 2, ..., m\} such that i<ji<j. (b) If n,m≥2n,m\ge 2 does there exist a polynomial QQ with real coefficients and degree nn such that: ∣Q(ai)−Q(aj)∣<∣ai−aj∣|Q(a_i)-Q(a_j)|<|a_i-a_j| for all i,j={1,2,...,m}i,j=\{1, 2, ..., m\} such that i<ji<j Edit: See #3
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