Tom has twelve slips of paper which he wants to put into five cups labeled $A$ , $B$ , $C$ , $D$ , $E$ . He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$ . The numbers on the papers are 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4 and 4.5. If a slip with 2 goes into cup $E$ and a slip with 3 goes into cup $B$ , then the slip with 3.5 must go into what cup?
A. $A$
B. $B$
C. $C$
D. $D$
E. $E$

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Worked solution

1. 2015 AMC 8 官方解析

总和 $= 2 \times 3 + 2.5 \times 2 + 3 \times 4 + 3.5 + 4.5 + 4 = 35$ ， $35 \div 5 = 7$ ，那么 $A$ 、 $B$ 、 $C$ 、 $D$ 、 $E$ 杯有和分别是 5、6、7、8、9。方法一： $5 = 2 + 3$ ， $6 = 3 + 3$ ；方法二： $5 = 2.5 + 2.5$ ， $6 = 3 + 3$ 。若方法二成立，3.5 和 4.5 必在一个瓶里， $3.5 + 4.5 = 8$ ，放在哪个瓶子里都不可能；若方法一成立，2.5 和 2.5 要分开，3.5 和 4.5 要分开，所以 $2.5 + 3.5 = 6$ ，另一个 2.5 与 4.5 一组， $2.5 + 4.5 = 7$ ，只能在 $E$ 瓶，所以 3.5 只能在 $D$ 瓶中。

Answer

D. $D$

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