AMC 8 2025 Problem 9
AMC8 2025 年第 9 题
统计与数据图表★★☆☆☆
Ningli looks at the 6 pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting 6 numbers?
Ningli观察时钟上6对正对着的数字，并计算每对数字的平均值。请问经过计算后得到的6个数的平均值是多少？
- A.5
- B.6.5
- C.8
- D.9.5
- E.12
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
This solution uses Gauss Summation. Our answer is 1 + 7 2 + 2 + 8 2 + ⋯ + 6 + 12 2 6 = 1 2 ( ( 1 + 7 ) + ( 2 + 8 ) + ⋯ + ( 6 + 12 ) ) 6 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 2 ⋅ 6 = 12 ⋅ 13 2 2 ⋅ 6 = 78 12 = (B) 6.5
We proceed with brute force. All of the pairs of opposite numbers on the clock are (12,6), (1,7), (2,8), (3,9), (4,10), (5,11), where order doesn't matter. The averages of each of these pairs are 9, 4, 5, 6, 7, and 8 respectively, and the average these numbers is . The problem is asking for the average of all 12 numbers. To find the average of all 12 numbers, you find the sum of all the integers from 1 to 12 which is 78, and divide it by 12 because there are 12 terms. Therefore, the answer is . You could also see that the average of, 2 and 8, is 5, only one more that it's consecutive pair, 1 and 7 which is 4. Assuming that this is the pattern, we can find the average 4 to 9 (6 consecutive integers starting with 4 ), and we that find our answer is 6.5.
继续练习