MathDB
Positive number

Source: Ukrainian Mathematical Olympiad 2021. Day 1, Problem 8.3

December 20, 2023
algebra

Problem Statement

The numbers (a1,a2,…,a2021)(a_1 , a_2 , \ldots, a_{2021}) and (b1,b2,…,b2021)(b_1 , b_2 , \ldots, b_{2021}) are some different permutations of the numbers (1,2,…,2021)(1, 2, \ldots, 2021), and the numbers (c1,c2,…,c2021)(c_1 , c_2 , \ldots, c_{2021}) are some permutation of the numbers (2,4,…,4042)(2, 4, \ldots, 4042). Prove that the given number D is positive:
D=c12−4a1b1a1+b1+c1+c22−4a2b2a2+b2+c2+…+c20212−4a2021b2021a2021+b2021+c2021D = \frac{c_1^2 -4 a_1b_1}{a_1 + b_1 + c_1} + \frac{c_2^2 - 4a_2b_2}{a_2 + b_2 + c_2} + \ldots + \frac{c_{2021}^2 - 4a_{2021}b_{2021}}{a_{2021} + b_{2021} + c_{2021}}
Proposed by Bogdan Rublov
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