Question

The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F - 32).$ 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 $T$ with $32 \le T \le 1000$ does the original temperature equal the final temperature?

答案

Solution:

Adding $9$ to $F$ adds exactly $5$ to $\frac{5}{9}(F - 32),$ hence $5$ to the rounded Celsius value, hence $9$ to the final Fahrenheit value. So $T$ returns to itself if and only if $T + 9$ does, and it suffices to check nine consecutive temperatures. Checking $32$ through $40:$ the final values are $32, 34, 34, 36, 36, 37, 37, 39, 39,$ so exactly the five temperatures $32, 34, 36, 37, 39$ survive.

The range from $32$ through $994$ contains $963 = 107 \cdot 9$ integers, contributing $107 \cdot 5 = 535$ survivors. The remaining $995, \ldots, 1000$ behave like $32, \ldots, 37,$ of which $32, 34, 36, 37$ survive, adding $4$ more.

The total is $535 + 4 = 539.$

2007 AIME I Problem 5

Solution:

Problem 5 in Other Years