The average of five numbers is $82$. The average of the two smallest numbers is $65$ and the average of the two greatest numbers is $100$. If all five numbers are arranged in increasing order, what is the middle number?

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**Solution:**

1. Let the five numbers in increasing order be $a_1, a_2, a_3, a_4, a_5$.

2. Given the average of all five numbers is $82$, so the sum of all five numbers is:
   $$	ext{Sum} = 5 	imes 82 = 410$$

3. The average of the two smallest numbers $a_1$ and $a_2$ is $65$, so their sum is:
   $$a_1 + a_2 = 2 	imes 65 = 130$$

4. The average of the two greatest numbers $a_4$ and $a_5$ is $100$, so their sum is:
   $$a_4 + a_5 = 2 	imes 100 = 200$$

5. Using the total sum, we find the middle number $a_3$:
   $$a_1 + a_2 + a_3 + a_4 + a_5 = 410$$
   Substitute the sums:
   $$130 + a_3 + 200 = 410$$
   $$a_3 = 410 - 330 = 80$$

**Answer:** $80$

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