Cho: Tính Q =

* Hướng dẫn giải

Từ $\frac{\text{x}}{\text{y\hspace{0.17em}+\hspace{0.17em}z\hspace{0.17em}+ t}}\text{=}\frac{\text{y}}{\text{x\hspace{0.17em}+\hspace{0.17em}t\hspace{0.17em}+\hspace{0.17em}z}}\text{=}\frac{\text{z}}{\text{x\hspace{0.17em}+\hspace{0.17em}y\hspace{0.17em}+\hspace{0.17em}t}}\text{=}\frac{\text{t}}{\text{x\hspace{0.17em}+\hspace{0.17em}y\hspace{0.17em}+\hspace{0.17em}z}}$

=> $\frac{\text{x}}{\text{y\hspace{0.17em}+\hspace{0.17em}z\hspace{0.17em}+\hspace{0.17em}t}}\text{+1=}\frac{\text{y}}{\text{x\hspace{0.17em}+\hspace{0.17em}t\hspace{0.17em}+\hspace{0.17em}z}}\text{+1=}\frac{\text{z}}{\text{x\hspace{0.17em}+ y\hspace{0.17em}+ t}}\text{+1=}\frac{\text{t}}{\text{x\hspace{0.17em}+\hspace{0.17em}y\hspace{0.17em}+ z}}+1$

=> $\frac{\text{x\hspace{0.17em}+ y + z + t}}{\text{y\hspace{0.17em}+ z + t}}\text{=}\frac{\text{y\hspace{0.17em}+ z + x + t}}{\text{x\hspace{0.17em}+ t + z}}\text{=}\frac{\text{z\hspace{0.17em}+\hspace{0.17em}x + t\hspace{0.17em}+\hspace{0.17em}y}}{\text{x\hspace{0.17em}+\hspace{0.17em}y\hspace{0.17em}+\hspace{0.17em}t}}\text{=}\frac{\text{t\hspace{0.17em}+\hspace{0.17em}x\hspace{0.17em}+\hspace{0.17em}y\hspace{0.17em}+\hspace{0.17em}z}}{\text{x\hspace{0.17em}+\hspace{0.17em}y\hspace{0.17em}+\hspace{0.17em}z}}$(*)

+) Nếu x + y + z + t = 0

=> x + y = –(z + t) ; y + z = – (x + t) ; z + t = – (x + y) ; t + x = – (y + z)

 Q = –1 –1 –1 –1 = –4           (0,5 điểm)

+) Nếu x + y + z + t  0 từ (*)  y + z + t = x + t + z = x + y + t = x + y + z

 x = y = t = z

Từ đó tính Q = 1 + 1 + 1 + 1 = 4         (0,5 điểm)