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CSMO Grade 11 Problem 5

Source: China Southeast Mathematical Olympiad

August 1, 2017
inequalities

Problem Statement

Let a,b,ca, b, c be real numbers, a≠0a \neq 0. If the equation 2ax2+bx+c=02ax^2 + bx + c = 0 has real root on the interval [−1,1][-1, 1]. Prove that min⁡{c,a+c+1}≤max⁡{∣b−a+1∣,∣b+a−1∣},\min \{c, a + c + 1\} \leq \max \{|b - a + 1|, |b + a - 1|\}, and determine the necessary and sufficient conditions of a,b,ca, b, c for the equality case to be achieved.
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