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Romanian District Olympiad 2019 - Grade 10 - Problem 4

Source: Romanian District Olympiad 2019 - Grade 10 - Problem 4

March 17, 2019
Inequalityalgebra

Problem Statement

Find the smallest positive real number λ\lambda such that for every numbers a1,a2,a3∈[0,12]a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right] and b1,b2,b3∈(0,∞)b_1,b_2,b_3 \in (0, \infty) with ∑i=13ai=∑i=13bi=1,\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1, we have b1b2b3≤λ(a1b1+a2b2+a3b3).b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3).
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