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Romanian District Olympiad 2019 - Grade 9 - Problem 1

Source: Romanian District Olympiad 2019 - Grade 9 - Problem 1

March 18, 2019
InequalityidentityCauchy Inequalityalgebra

Problem Statement

Let n∈N,n≥2n \in \mathbb{N}, n \ge 2 and the positive real numbers a1,a2,…,ana_1,a_2,…,a_n and b1,b2,…,bnb_1,b_2,…,b_n such that a1+a2+…+an=b1+b2+…+bn=S.a_1+a_2+…+a_n=b_1+b_2+…+b_n=S. <spanclass=′latex−bold′>a)</span><span class='latex-bold'>a)</span> Prove that ∑k=1nak2ak+bk≥S2.\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k} \ge \frac{S}{2}. <spanclass=′latex−bold′>b)</span><span class='latex-bold'>b)</span> Prove that ∑k=1nak2ak+bk=∑k=1nbk2ak+bk.\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k}= \sum\limits_{k=1}^n \frac{b_k^2}{a_k+b_k}.
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