A2. For how many of the integers from 10 to 99 is the sum of the digits equal to the square of an integer?

(An example is the number 27 with the sum of the digits $2+7=9=3^{2}$.)
(A) 13
(B) 14
(C) 15
(D) 16
(E) 17
A2. For how many of the integers from 10 to 99 is the sum of the digits equal to the square of an integer? (An example is the number 27 with the sum of the digits $2+7=9=3^{2}$.) (E) 17

Solution We look at how many of the numbers have a digit sum of 1, 2, etc. There is 1 number with sum 1 (namely 10); there are 2 with sum 2 (namely 20 and 11); etc.; 9 with sum 9 (namely 90, 81, ..., 18); also 9 with sum 10 (namely 91, 82, ..., 19); etc. up to 1 with sum 18 (namely 99); see table. The sum is a square (i.e., 1, 4, 9, or 16) for $1+4+9+3=17$ of the 90 numbers.

| sum: | $\mathbf{1}$ | 2 | 3 | $\mathbf{4}$ | 5 | 6 | 7 | 8 | $\mathbf{9}$ | 10 | 11 | 12 | 13 | 14 | 15 | $\mathbf{1 6}$ | 17 | 18 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| number: | $\mathbf{1}$ | 2 | 3 | $\mathbf{4}$ | 5 | 6 | 7 | 8 | $\mathbf{9}$ | 9 | 8 | 7 | 6 | 5 | 4 | $\mathbf{3}$ | 2 | 1 |