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, 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?
I have no idea how to start on this problem. I tried putting in some numbers to see if that got some stuff rolling, but that didn't work at all. Any ideas?
The outcome of the first rounding step depends on the starting temperature's remainder modulo $9$, so it's reasonable to write the initial temperature as $9q+r$ for some $(q,r)$ and see what happens for different values of $r$. For example, we have:
$$\frac59(9q+0 - 32) = 5q - \frac{160}{9} \leadsto 5q - 18$$
where I denote by $\leadsto$ the result of rounding. Then we can do the reverse calculation:
$$\frac95(5q-18) + 32 = 9q - 0.6 \leadsto 9q - 1.$$
So we see that when $r=0$, we don't get the same number back. (If this is enough of a hint, stop reading here and try to solve the problem on your own.)
We can do the same calculation for other values of $r$. For $r=1$, we have
$$(9q + 1) ^\circ F = \left(5q - \frac{155}{9}\right)^\circ C \leadsto (5q-17)^\circ C = (9q + 1.2)^\circ F \leadsto (9q+1)^\circ F$$ so we do get the original number back.
Given 3 hours to do 15 problems, it is not unreasonable to take the time to just continue on in this way for $r=2,3,4,5,6,7,8$. We conclude that $r=1,3,5,7,8$ give back the original number, and the rest don't.
Starting at a temperature between $32$ and $1000$ means starting at $9\cdot 3 + 5$ and ending at $9\cdot 111 + 1$. So:
The final answer is therefore $539$.