AIME 2026 II · 第 1 题

AIME 2026 II — Problem 1

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem

Find the sum of the $10$ th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms.

解析

Solution 1

Suppose we have an arithmetic sequence of integers that includes $4, 24,$ and $34$ . The common difference of the sequence $d$ , must satisfy

$d\mid 24-4 \text{ and } d \mid 34-4 \implies d \mid \text{gcd}(24-4, 34-4) = 10$ Since $d$ must be positive, $d=1, 2, 5, 10$ . The 10th terms of each of these sequences is $13,22,49,94$ , respectively, so the sum is $\boxed{178}$ . ~sillybone