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7

Part of 2023 South East Mathematical Olympiad

Problems(1)

min max \sum a_ib_i if sum (a_i^2-b_1^2)=1 and sum (a_i+b_i)=2023

Source: 2023 China South-east Mathematical Olympiad Grade 10 P7 CSMO

4/6/2024
The positive integer number SS is called a "line number". if there is a positive integer nn and 2n2n positive integers a1a_1, a2a_2,...,ana_n, b1b_1,b2b_2,...,bnb_n, such that S=∑i=1naibiS = \sum^n_{i=1} a_ib_i, ∑i=1n(ai2−b12)=1\sum^n_{i=1} (a_i^2-b_1^2)=1, and ∑i=1n(ai+bi)=2023\sum^n_{i=1} (a_i+b_i)=2023, find: (1) The minimum value of line numbers. (2)The maximum value of line numbers.
algebrainequalities