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Problem 9. For real number a, let bac denote the largest integer less than or equal to a, and let {a}, the fractional part of a, be defined by {a} = a − bac. As examples, b3.6c = 3, {3.6} = 0.6, b−3.6c = −4, and {−3.6} = 0.4. Find all real number solutions (x, y, z) to the system x + byc + {z} bxc + {y} + z {x} + y + bzc = − 7.1 = 18.8 = 4.7. Solution. We use the fact that for any real number a, a = bac + {a}, which is equivalent to a − {a} = bac. In the original system of equation. add the first two equation and subtract the third, keeping in mind the relations just mentioned: (x + bxc − {x}) + (byc + {y} − y) + (z + z − bzc) = (−7.1 + 18.8 − 4.7), leading to 7 = 3.5. 2 Because bxc is an integer and {z} is a positive number in [0, 1), it follows that bxc = 3 and {z} = 0.5. bxc + {z} = By similar reasoning, adding the first and third of the above three equation, then subtracting the second equation leads to {x}+byc = −10.6 = −11+0.4 from which {x} = 0.4 and byc = −11. Adding the second and third equations and subtracting the first we find {y} + bzc = 15.3 from which {y} = 0.3 and bzc = 15. We can now recover x, y, z by using a = bac + {a} to see x = 3.4, y = −11 + 0.3 = −10.7 z = 15.5