Difference between revisions of "2007 AIME II Problems/Problem 6"
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| − | == Problem == | + | <noinclude>== Problem == |
| − | An integer is called ''parity-monotonic'' if its decimal representation <math>a_{1}a_{2}a_{3}\cdots a_{k}</math> satisfies <math>a_{i}<a_{i+1}</math> if <math>a_{i}</math> is [[odd]], and <math>a_{i}>a_{i+1}</math> if <math>a_{i}</math> is [[even]]. How many four-digit parity-monotonic integers are there? | + | </noinclude>An integer is called ''parity-monotonic'' if its decimal representation <math>a_{1}a_{2}a_{3}\cdots a_{k}</math> satisfies <math>a_{i}<a_{i+1}</math> if <math>a_{i}</math> is [[odd]], and <math>a_{i}>a_{i+1}</math> if <math>a_{i}</math> is [[even]]. How many four-digit parity-monotonic integers are there?<noinclude> |
== Solution == | == Solution == | ||
Let's set up a table of values. Notice that 0 and 9 both cannot appear as any of <math>a_1,\ a_2,\ a_3</math> because of the given conditions. A clear pattern emerges. | Let's set up a table of values. Notice that 0 and 9 both cannot appear as any of <math>a_1,\ a_2,\ a_3</math> because of the given conditions. A clear pattern emerges. | ||
| + | |||
| + | For example, for <math>3</math> in the second column, we note that <math>3</math> is less than <math>4,6,8</math>, but greater than <math>1</math>, so there are four possible places to align <math>3</math> as the second digit. | ||
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
| + | </noinclude> | ||
Revision as of 15:37, 30 March 2007
Problem
An integer is called parity-monotonic if its decimal representation 
satisfies 
if 
is odd, and 
if 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 
because of the given conditions. A clear pattern emerges.
For example, for 
in the second column, we note that is less than 
, but greater than 
, so there are four possible places to align as the second digit.
| Number | 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 
.
See also
| 2007 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
