Question

[Lê Trần Gia Bảo]

Cho a,b,c thuộc R thoả mãn

ab+bc+ca=abc và a+b+c=1

tìm a,b,c

Answer 1

[Nguyễn Lê Phước Thịnh]

Ta có: ab+bc+ca=abc

nên abc-ab-bc-ac=0

Ta có: a+b+c=1

nên a+b+c-1=0

Ta có: abc-ab-bc-ac+a+b+c-1=0

\(\Leftrightarrow\left(abc-ab\right)-\left(bc-b\right)-\left(ac-a\right)+\left(c-1\right)=0\)

\(\Leftrightarrow ab\left(b-1\right)-b\left(c-1\right)-a\left(c-1\right)+\left(c-1\right)=0\)

\(\Leftrightarrow b\left(c-1\right)\left(a-1\right)-\left(c-1\right)\left(a-1\right)=0\)

\(\Leftrightarrow\left(a-1\right)\left(b-1\right)\left(c-1\right)=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=1\\c=1\end{matrix}\right.\)