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United States competition problems

Number theory Difficulty 5.3 AIME, harder Prove it United States

Problem:

January 3, 1911 was an odd date as its abbreviated representation, $1 / 3 / 1911$ , can be written using only odd digits (note all four digits are written for the year). To the nearest month, how many months will have elapsed between the most recent odd date and the next odd date (today is $3 / 3 / 2001$ , an even date).

Solution

Solution:

The most recent odd date was $11 / 19 / 1999$ (November has 30 days, but the assumption that it has 31 days does not change the answer), and the next odd date will be $1 / 1 / 3111$ . From $11 / 19 / 1999$ to $1 / 1 / 2000$ is about 1 month. From 2000 to 3111 is 1111 years, or $12 \cdot 1111 = 13332$ months, so the total number of months is $13333$ .

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