Five distinct integers from 1 to 10 are chosen, and five distinct integers from 11 to 20 are chosen. No two numbers differ by exactly 10. What is the sum of the ten chosen numbers?

A. 95

B. 100

C. 105

D. 110

E. 115

1. 2025 AMC 8 官方解析

Answer (C): List the numbers 1 through 20 in a table:

No two of the chosen numbers differ by 10, so only one number can be chosen from each column of the table. Since there are 10 columns and 10 numbers are chosen, exactly one number comes from each column of the table with five numbers chosen from the top row and five numbers chosen from the bottom row.

Any number chosen from the bottom row will contribute 10 more to the total sum than its counterpart in the top row. This means the sum can be broken down as the sum of all the numbers in the top row, together with an additional 10 for each number chosen from the bottom row. Therefore, the sum of the ten chosen numbers is
$(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) + 5 \cdot 10 = 55 + 50 = 105.$

As an example, suppose the numbers chosen in the top row were 1, 3, 5, 7, and 9, and the numbers chosen from the bottom row were 12, 14, 16, 18, and 20. Their sum is
$1 + 12 + 3 + 14 + 5 + 16 + 7 + 18 + 9 + 20$
$= 1 + (2 + 10) + 3 + (4 + 10) + 5 + (6 + 10) + 7 + (8 + 10) + 9 + (10 + 10)$
$= (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) + 5 \cdot 10$
$= 105,$

as shown above.