AMC 8 2020 Problem 16
AMC8 2020 年第 16 题
基本计数原理★★★☆☆
Each of the points A,B,C,D,E, and F in the figure below represents a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47. What is the digit represented by B?
下图中 A，B，C，D，E，F 中的每个点代表 1 到 6 之间的一个不同的数字。图中所示的 5 条直线，每条都会通过其中几个点。每条直线上的几个数字相加得到 5 个和，每条直线一个和。把这 5 个和相加，结果为 47，那么 B 代表的数字是多少？
- A.1
- B.2
- C.3
- D.4
- E.5
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
We can form the following expressions for the sum along each line:
$\begin{dcases}A+B+C\\A+E+F\\C+D+E\\B+D\\B+F\end{dcases}
$ Adding these together, we must have $2A+3B+2C+2D+2E+2F=47$, i.e. $2(A+B+C+D+E+F)+B=47$. Since A,B,C,D,E,F are unique integers between 1 and 6, we obtain $A+B+C+D+E+F=1+2+3+4+5+6=21$ (where the order doesn't matter as addition is commutative), so our equation simplifies to $42 + B = 47$. This means $B = 5$.Following the first few steps of, we have 2(A+C+D+E+F)+3B=47. Because an even number 2(A+C+D+E+F) subtracted from an odd number (47) is always odd, we know that is odd, showing that is odd. Now we know that is 1, 3, or 5. If we try B=1, we get 43=47, which is not true. Also when we test B=3, we get 45=47, which is also false. Therefore, we end up getting B = 5. -minor edit by tonykuncheng -minor edit by Lebron James
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