2025 AMC 8

Complete problem set with solutions and individual problem pages

Problem 16 Hard

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

Answer:C

Solution 1

Note that for no two numbers to differ by , every number chosen must have a different units digit. To make computations easier, we can choose from the first group and from the second group. Then the sum evaluates to .

Solution 2

For , we can add the first term and the last term, which is . If we add the second term and the second-to-last term it is also . There are pairs that sum to , so the answer is which equals .