Problem:
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?
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Solution:
The monotonous positive integers with one digit or increasing digits can be put into a one-to-one correspondence with the nonempty subsets of . The number of such subsets is . The monotonous positive integers with one digit or decreasing digits can be put into a one-to-one correspondence with the subsets of other than and . The number of these is . The single-digit numbers are included in both sets, so there are monotonous positive integers.
The problems on this page are the property of the MAA's American Mathematics Competitions