2007 AIME I Problems/Problem 5
Contents
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 if for some integer . 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 .
Note
Proof that iff for some integer :
First assume that cannot be written in the form for any integer . Let . Our equation simplifies to . However, this equation is not possible, as we defined such that it could not be written in this form. Therefore, if , then .
Now we will prove that if , . We realize that because of the 5 in the denominator of , will be at most away from . Let , meaning that . Now we substitute this into our equation:
.
Now we use the fact that
Hence and we are done.
- mako17
Solution 3
Let be a degree Celsius, and rounded to the nearest integer. Since was rounded to the nearest integer we have , which is equivalent to if we multiply by . Therefore, it must round to because so is the closest integer. Therefore there is one solution per degree celcius in the range from to , meaning there are solutions.
Solution 4
Start listing out values for and their corresponding values of . You will soon find that every 9 values starting from = 32, there is a pattern:
: Works
: Doesn't work
: work
: Doesn’t work
: Works
: Works
: Doesn’t work
: Works
: Doesn’t work
: Works
There are numbers between and , inclusive. This is sets of , plus extra numbers at the end. In each set of , there are “Works,” so we have values of that work.
Now we must add the extra numbers. The number of “Works” in the first terms of the pattern is , so our final answer is solutions that work.
Submitted by warriorcats
Solution 5(similar to solution 3 but faster solution if you have no time)
Notice that every value corresponds to exactly one value but multiple values can correspond to a value. Thus, the smallest value is and the largest value is yielding solutions.
-alanisawesome2018
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 |
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