Question

Cho a,b,c thuộc R, a² + b² + c² = 3 và a + b + c + ab + ac + bc = 6
Tính giá trị biểu thức \(Q=\frac{a^{22}+b^{12}+c^{1994}}{a^{22}+b^{12}+c^{2013}}\)

Answer 1

Ta có:

\(a+b+c+ab+bc+ca=6\)

\(\Leftrightarrow12-\left(2a+2b+2c+2ab+2bc+2ca\right)=0\)

\(\Leftrightarrow3\left(a^2+b^2+c^2\right)+3-\left(2a+2b+2c+2ab+2bc+2ca\right)=0\)

\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)+\left(a^2-2a+1\right)+\left(b^2-2b+1\right)+\left(c^2-2c+1\right)=0\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2+\left(a-1\right)^2+\left(b-1\right)^2+\left(c-1\right)^2=0\)

\(\Rightarrow a=b=c=1\)

\(\Rightarrow Q=\frac{1^{22}+1^{12}+1^{1994}}{1^{22}+1^{12}+1^{2013}}=\frac{3}{3}=1\)

Answer 2

vào máy tính bấm sẽ ra đáp án = 1