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Polynomial mixed with Inequality

Source: Iranian MO 2009 2nd round - IMO Longlist 1982 P56

April 7, 2010
algebrapolynomialinequalitiesconicsparabolaanalytic geometrygraphing lines

Problem Statement

Let f(x)=ax2+bx+cf(x) = ax^2 + bx+ c and g(x)=cx2+bx+ag(x) = cx^2 + bx + a. If ∣f(0)∣≤1,∣f(1)∣≤1,∣f(−1)∣≤1|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1, prove that for ∣x∣≤1|x| \leq 1,
(a) ∣f(x)∣≤5/4|f(x)| \leq 5/4,
(b) ∣g(x)∣≤2|g(x)| \leq 2.
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