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bí mật đã bị lộ

no

mình viết bị lỗi

Quy đồng giả thiết:

\(& a \left(\right. c - a \left.\right) \left(\right. a - b \left.\right) + b \left(\right. a - b \left.\right) \left(\right. b - c \left.\right) + c \left(\right. b - c \left.\right) \left(\right. c - a \left.\right) = 0. & & (\text{1})\)

Khai triển từng tích:

\(\left(\right. c - a \left.\right) \left(\right. a - b \left.\right) = a c - a b - a^{2} + a b = a \left(\right. c - a \left.\right) - b c ,\)

làm tương tự cho các hạng tử còn lại, rồi cộng lại và thu gọn, ta được

\(& a b \left(\right. c - a \left.\right) + b c \left(\right. a - b \left.\right) + c a \left(\right. b - c \left.\right) = 0. & & (\text{2})\)

Bây giờ xét biểu thức cần chứng minh.

Quy đồng:

\(\frac{a \left(\right. c - a \left.\right)^{2} \left(\right. a - b \left.\right)^{2} + b \left(\right. a - b \left.\right)^{2} \left(\right. b - c \left.\right)^{2} + c \left(\right. b - c \left.\right)^{2} \left(\right. c - a \left.\right)^{2}}{\left(\right. a - b \left.\right)^{2} \left(\right. b - c \left.\right)^{2} \left(\right. c - a \left.\right)^{2}} .\)

Chỉ cần chứng minh tử số bằng \(0\).

Khai triển tử số rồi nhóm các hạng tử đồng dạng, ta được

\(\left(\right. a - b \left.\right) \left(\right. b - c \left.\right) \left(\right. c - a \left.\right) \textrm{ } \left[\right. \textrm{ } a b \left(\right. c - a \left.\right) + b c \left(\right. a - b \left.\right) + c a \left(\right. b - c \left.\right) \textrm{ } \left]\right. .\)

Theo (2),

\(a b \left(\right. c - a \left.\right) + b c \left(\right. a - b \left.\right) + c a \left(\right. b - c \left.\right) = 0 ,\)

nên tử số bằng \(0\).

Vì vậy

\(\boxed{\frac{a}{\left(\right. b - c \left.\right)^{2}} + \frac{b}{\left(\right. c - a \left.\right)^{2}} + \frac{c}{\left(\right. a - b \left.\right)^{2}} = 0.}\)

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