If x₁x₂x₃ = 4(4 + x₁ + x₂ + x₃), then what is the value of $$\left[ {\frac{1}{{2 + {x_1}}}} \right] + \left[ {\frac{1}{{2 + {x_2}}}} \right] + \left[ {\frac{1}{{2 + {x_3}}}} \right]?$$

Solution (By Examveda Team)

$$\eqalign{ & {x_1}{x_2}{x_3} = 4\left( {4 + {x_1} + {x_2} + {x_3}} \right) \cr & \left[ {\frac{1}{{2 + {x_1}}}} \right] + \left[ {\frac{1}{{2 + {x_2}}}} \right] + \left[ {\frac{1}{{2 + {x_3}}}} \right] = ? \cr & {\text{Assume value of }}{x_1},\,{x_2}\,\& \,{x_3} \cr & {x_1} = 4,\,{x_2} = 4,\,{x_3} = 4 \cr & 4 \times 4 \times 4 = 4\left( {4 + 4 + 4 + 4} \right) \cr & 64 = 64{\text{ value satisfied}} \cr & \therefore \left[ {\frac{1}{{2 + 4}} + \frac{1}{{2 + 4}} + \frac{1}{{2 + 4}}} \right] \cr & = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \cr & = \frac{3}{6} \cr & = \frac{1}{2} \cr} $$