Difference between revisions of "2007 AIME I Problems/Problem 5"
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=== Solution 2 === | === Solution 2 === | ||
− | + | Notice that <math>\left[ \frac{9}{5} \left[ \frac{5}{9} k \right] \right] = k</math> holds iff <math>k=\left[ \frac{9}{5}x\right]</math> for some <math>x</math>. | |
+ | Thus, after translating from <math>F\to F-32</math> we want count how many values of <math>x</math> there are such that <math>k=\left[ \frac{9}{5}x\right]</math> is an integer from <math>0</math> to <math>968</math>. This value is computed as <math>\left[968*\frac{5}{9}\right]+1</math>, adding in the extra solution corresponding to <math>0</math>. | ||
=== Solution 3 === | === Solution 3 === |
Revision as of 20:54, 15 March 2007
Problem
The formula for converting a Fahrenheit temperature to the corresponding Celsius temperature is An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.
For how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature?
Solution
Solution 1
Examine modulo 9.
- If , then we can define . This shows that . This case works.
- If , then we can define . This shows that . So this case doesn't work.
Generalizing this, we define that . Thus, . We need to find all values that . Testing every value of shows that , so of every values of work.
There are cycles of , giving numbers that work. Of the remaining numbers from onwards, work, giving us as the solution.
Solution 2
Notice that holds iff for some . Thus, after translating from we want count how many values of there are such that is an integer from to . This value is computed as , adding in the extra solution corresponding to .
Solution 3
Let be a degree Celcius, and rounded to the nearest integer. so it must round to because . Therefore there is one solution per degree celcius in the range from to , meaning there are solutions.
See also
2007 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |