Paulinho is training for his arithmetic test. To become faster, he keeps performing various additions. Paulinho asks his father to help by choosing five integers from 1 to 20. Then, Paulinho adds them up. After several attempts, his father noticed that he was getting very bored with the additions and decided to make his son think more before answering. Let's help Paulinho answer his father's new questions?

(a) What is the smallest and the largest possible sum of five integers from 1 to 20?

(b) For each value from the smallest to the largest possible, select five numbers from 1 to 20 such that their sum is this value.
Solution

(a) The minimum sum is obtained when the five smallest numbers are used, that is, the smallest possible sum is $1+2+3+4+5=15$. Using the same reasoning, the largest possible sum is $16+17+18+19+20=90$.
(b) Yes, let's prove that we can increment the value of the sum of five numbers by 1 starting from 15 until we reach 90. We start from the minimum sum $1+2+3+4+5$. Increase the largest number that has not yet reached its maximum value by 1. In other words, increase the 5 to 20. Then, increase the 4 to 19. In this way, we have:

$$
\begin{aligned}
1+2+3+4+5 & =15 \\
1+2+3+4+6 & =16 \\
1+2+3+4+7 & =17 \\
& \cdots \\
1+2+3+4+20 & =30 \\
1+2+3+5+20 & =31 \\
1+2+3+6+20 & =32 \\
& \cdots \\
1+2+3+19+20 & =45 \\
& \cdots \\
15+17+18+19+20 & =89 \\
16+17+18+19+20 & =90
\end{aligned}
$$

Notice that each number from 15 to 90 will appear as the sum of five distinct numbers.