Problem:
The formula for converting a Fahrenheit temperature to the corresponding Celsius temperature is . An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer. For how many integer Fahrenheit temperatures with does the original temperature equal the final temperature?
Solution:
Note that a temperature converts back to if and only if converts back to . Thus it is only necessary to examine nine consecutive temperatures. It is easy to show that 32 converts back to and both convert back to and both convert back to and both convert back to , and and both convert back to . Hence out of every nine consecutive temperatures starting with , five are converted correctly and four are not. For . There are temperatures that are converted correctly. The remaining six temperatures behave like , so four of the remaining six temperatures are converted correctly. Thus there is a total of temperatures.
Because one Fahrenheit degree is of a Celsius degree, every integer Celsius temperature is the conversion of either one or two Fahrenheit temperatures (nine Fahrenheit temperatures are being converted to only five Celsius temperatures) and converts back to one of those temperatures. The Fahrenheit temperatures and convert to and , respectively, which convert back to and . Therefore there are Fahrenheit temperatures with the required property, corresponding to the integer Celsius temperatures from to .
The problems on this page are the property of the MAA's American Mathematics Competitions