AMC8 2015

AMC8 2015 · Q23

AMC8 2015 · Q23. It mainly tests Fractions, Casework.

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?

Tom有十二张纸条，他想放入标有A、B、C、D、E的五个杯子中。他希望每个杯子中纸条数字之和为整数。而且，他希望这五个整数从A到E连续递增。纸条上的数字是2、2、2、2.5、2.5、3、3、3、3、3.5、4和4.5。如果一张2的纸条放入杯子E，一张3的纸条放入杯子B，那么3.5的纸条必须放入哪个杯子？

(A) A A

(B) B B

(D) D D

(E) E E

Answer

Correct choice: (D)

正确答案：（D）

Solution

The sum of the numbers is $35$. So the $5$ consecutive numbers in the cups must be $5$, $6$, $7$, $8$, and $9$. It is impossible to get a sum of $5$ or $7$ using the slip with $3.5$. Cup $B$ needs a sum of $6$, but it already has a slip with $3$ on it so the slip with a $3.5$ can’t go there. Cup $E$ needs a sum of $9$, but with a slip with $2$ in it the slip with $3.5$ can’t go there. The only place the slip with $3.5$ on it can go is Cup $D$. One possibility is:

这些数的和是 $35$。因此杯子里的 $5$ 个连续整数必须是 $5$、$6$、$7$、$8$、$9$。使用写着 $3.5$ 的纸条，不可能凑出和为 $5$ 或 $7$。杯子 $B$ 需要和为 $6$，但它已经有一张写着 $3$ 的纸条，所以写着 $3.5$ 的纸条不能放在那里。杯子 $E$ 需要和为 $9$，但它里面有一张写着 $2$ 的纸条，因此写着 $3.5$ 的纸条也不能放在那里。写着 $3.5$ 的纸条唯一能放的地方是杯子 $D$。一种可能的安排是：

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