Question

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Tìm x , biết :

| x + \(\dfrac{1}{101}|\) + | x + \(\dfrac{2}{101}|\) + | x + \(\dfrac{3}{101}|\) + ...... + | x + \(\dfrac{100}{101}\) | = 101x

Answer 1

[Nguyễn Lê Phước Thịnh]

Vì \(\left|x+\frac{1}{101}\right|+\left|x+\frac{2}{101}\right|+\left|x+\frac{3}{101}\right|+...+\left|x+\frac{100}{101}\right|>0\forall x\)

mà \(\left|x+\frac{1}{101}\right|+\left|x+\frac{2}{101}\right|+\left|x+\frac{3}{101}\right|+...+\left|x+\frac{100}{101}\right|=101x\)

nên x>0

Với x>0, ta được:

\(x+\frac{1}{101}+x+\frac{2}{101}+x+\frac{3}{101}+...+x+\frac{100}{101}=101x\)

\(\Leftrightarrow100x-101x+\frac{5050}{101}=0\)

\(\Leftrightarrow-x+50=0\)

hay x=50

Vậy: S={50}