|x−1|+|x−2|+|x−3|+...+|x−2020||x−1|+|x−2|+|x−3|+...+|x−2020|
=(|x−1|+|2020−x|)+(|x−2|+|2019−x|)+...+(|x−1010|+|1011−x|)=(|x−1|+|2020−x|)+(|x−2|+|2019−x|)+...+(|x−1010|+|1011−x|)
≥|x−1+2020−x|+|x−2+2019−x|+...+|x−1010+1011−x|≥|x−1+2020−x|+|x−2+2019−x|+...+|x−1010+1011−x|
=2019+2017+...+1=2019+2017+...+1
=(2019+1).[(2019−1)÷2+1]2=1020100=(2019+1).[(2019−1)÷2+1]2=1020100
Dấu ==khi \hept⎧⎨⎩(x−1)(2020−x)≥0...(x−1010)(1011−x)≥0⇔1010≤x≤1011\hept{(x−1)(2020−x)≥0...(x−1010)(1011−x)≥0⇔1010≤x≤1011.
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