Đáp án: $A\ge1020100$
Giải thích các bước giải:
Ta có:
$A=|x-1|+|x-2|+|x-3|+...+|x-2020|$
$\to A=(|x-1|+|x-2020|)+(|x-2|+|x-2019|)+...+(|x-1009|+|x-1010|)$
$\to A=(|x-1|+|2020-x|)+(|x-2|+|2019-x|)+...+(|x-1009|+|1010-x|)$
$\to A\ge |x-1+2020-x|+|x-2+2019-x|+..+|x-1009+1010-x|$
$\to A\ge 2019+2017+2015+...+1$
$\to A\ge \dfrac{(2019+1)\cdot 1010}{2}=1020100$
Dấu = xảy ra khi:
$(x-1)(2020-x)\ge 0, (x-2)(2019-x)\ge 0,.. , (x-1009)(1010)-x\ge 0$
$\to 1\le x\le 2020, 2\le x\le 2019, .., 1009\le x\le 1010$
$\to 1009\le x\le 1010$
