Problem:
How many four-digit positive integers , with , have the property that the three two-digit integers form an increasing arithmetic sequence? One such number is , where , and .
Answer Choices:
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Solution:
Let be the common difference for the arithmetic sequence. If or , then must be a multiple of , so . However, the two-digit integers and are then equal, a contradiction. Therefore either or is an increasing arithmetic sequence.
Case 1: is an increasing arithmetic sequence. In this case the additions of to and do not involve any carries, so also forms an increasing arithmetic sequence, as does . Let . If , the possible values of are , and . If , the possible values of are , and . There are no possibilities with . Thus in this case there are integers that have the required property: , , and .
Case 2: is an increasing arithmetic sequence. In this case the addition of to involves a carry, so forms a nondecreasing arithmetic sequence, as does . Hence is a nondecreasing arithmetic sequence. Again letting , note that and . The only integers with the required properties are with and with , and with ; and and with . Thus in this case there are integers that have the required property.
The total number of integers with the required property is .
The problems on this page are the property of the MAA's American Mathematics Competitions