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min sum a_i^2/(a_i+b_i) when sum a_i=1, sum b_i=1

Source: 2019 Argentina OMA Finals L3 p2

January 16, 2023

algebraSuminequalities

Problem Statement

Let

n\geq1

be an integer. We have two sequences, each of

n

positive real numbers

a_1,a_2,\ldots ,a_n

and

b_1,b_2,\ldots ,b_n

such that

a_1+a_2+\ldots +a_n=1

and

b_1+b_2+\ldots +b_n=1

. Find the smallest possible value that the sum can take

\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+\ldots +\frac{a_n^2}{a_n +b_n}.