Two sets each contain 3 consecutive positive integers

题目

Two sets each contain 3 consecutive positive integers. None of the integers included in one set are included in the other set. The two sets are combined to form a set of 6 ordered positive integers. If the median of the combined set is also an integer, which of the following could be the range of the integers in the new set?

I. 15

II. 16

III. 17

选项

I only

II only

III only

I and III only

I, II, and III

解析

设第一个集合 = \(\{(x-1), x, (x+1)\}\) 第二个集合 = \(\{(y-1), y, (y+1)\}\) 两个集合的平均数 = \(\frac{3x + 3y}{6}\) 根据题目，\((x + y)\)是2的倍数，因此\(x\)和\(y\)要么都是偶数，要么都是奇数。两个集合合并后的极差 = \(|x - y| + 2\) 从上述表达式可知，无论\(x\)、\(y\)是同偶还是同奇，这个结果始终是偶数。答案：B

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