2007 AIME II Problems/Problem 6

Revision as of 13:33, 21 December 2011 by Baijiangchen (talk | contribs) (Solution)

Problem

An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there?

Solution

Let's set up a table of values. Notice that 0 and 9 both cannot appear as any of $a_1,\ a_2,\ a_3$ because of the given conditions. A clear pattern emerges.

For example, for $3$ in the second column, we note that $3$ is less than $4,6,8$, but greater than $1$, so there are four possible places to align $3$ as the second digit.

Digit   1st   2nd   3rd   4th
0 0 0 0 64
1 1 4 16 64
2 1 4 16 64
3 1 4 16 64
4 1 4 16 64
5 1 4 16 64
6 1 4 16 64
7 1 4 16 64
8 1 4 16 64
9 0 0 0 64

For any number from 1-8, there are exactly 4 numbers from 1-8 that are odd and less than the number or that are even and greater than the number (the same will happen for 0 and 9 in the last column). Thus, the answer is $4^{k-1} \cdot 10 = 4^3\cdot10 = 640$.

See also

2007 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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All AIME Problems and Solutions
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