Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap of the track in a time, $t$ seconds, where $t$ is a positive integer. The product of the smallest and largest possible integer values of $t$ is
(A) 3705
(B) 3762
(C) 2816
(D) 3640
(E) 3696
The two joggers meet every 36 seconds.

Therefore, the combined distance that the two joggers run every 36 seconds is equal to the total distance around one lap of the oval track, which is constant.

Thus the greater the first jogger's constant speed, the greater the distance that they run every 36 seconds, meaning the second jogger runs less distance in the same time (their combined
distance is constant) and hence the smaller the second jogger's constant speed.

Conversely, the slower the first jogger's constant speed, the less distance that they run every 36 seconds, meaning the second jogger must run a greater distance in this same time and hence the greater the second jogger's constant speed.

This tells us that if the first jogger completes one lap of the track as fast as possible, which is in 80 seconds, then the second jogger's time to complete one lap of the track is as slow as possible.

We will call this time $t_{\text {max }}$, the maximum possible time that it takes the second jogger to complete one lap of the track.

Similarly, if the first jogger completes one lap of the track as slowly as possible, which is in 100 seconds, then the second jogger's time to complete one lap of the track is as fast as possible.

We will call this time $t_{\text {min }}$, the minimum possible time that it takes the second jogger to complete one lap of the track.

## Finding the value of $t_{\max }$

Recall that $t_{\max }$ is the time it takes the second jogger to complete one lap when the first jogger completes one lap of the track in 80 seconds.

If the first jogger can complete one lap of the track in 80 seconds, then in 36 seconds of running, the first jogger will complete $\frac{36}{80}=\frac{9}{20}$ of a complete lap of the track.

In this same 36 seconds, the two joggers combined distance running is 1 lap, and so the second jogger runs $1-\frac{9}{20}=\frac{11}{20}$ of a complete lap.

If the second jogger runs $\frac{11}{20}$ of a complete lap in 36 seconds, then the second jogger runs $\frac{20}{11} \times \frac{11}{20}=1$ complete lap in $\frac{20}{11} \times 36=\frac{720}{11}$ seconds. Thus, $t_{\max }=\frac{720}{11}=65 . \overline{45}$ seconds.

Finding the value of $t_{\min }$

Recall that $t_{\min }$ is the time it takes the second jogger to complete one lap when the first jogger completes one lap of the track in 100 seconds.

If the first jogger can complete one lap of the track in 100 seconds, then in 36 seconds of running, the first jogger will complete $\frac{36}{100}=\frac{9}{25}$ of a complete lap of the track.

In this same 36 seconds, the two joggers combined distance running is 1 lap, and so the second jogger runs $1-\frac{9}{25}=\frac{16}{25}$ of a complete lap.

If the second jogger runs $\frac{16}{25}$ of a complete lap in 36 seconds, then the second jogger runs $\frac{25}{16} \times \frac{16}{25}=1$ complete lap in $\frac{25}{16} \times 36=\frac{900}{16}$ seconds. Thus, $t_{\min }=\frac{900}{16}=56.25$ seconds.

Determining the product of the smallest and largest integer values of $t$

Since the second jogger completes 1 lap of the track in at most $65 . \overline{45}$ seconds, then the largest possible integer value of $t$ is 65 seconds.

The second jogger completes 1 lap of the track in at least 56.25 seconds, so then the smallest possible integer value of $t$ is 57 seconds.

Finally, the product of the smallest and largest integer values of $t$ is $57 \times 65=3705$.