A number is said to be balanced if one of its digits is the arithmetic mean of the others. For example, 132, 246, and 777 are balanced. How many balanced three-digit numbers exist?
Note that if a balanced number has three distinct non-zero digits, then with the same digits, we obtain six balanced numbers. For this, it is enough to swap the digits' positions. For example, 123, 132, 213, 231, 312, and 321.

If one of the three digits of the balanced number is 0, then with these digits, we obtain only four balanced numbers, as 0 cannot be in the hundreds place. For example, 102, 120, 201, and 210.

Thus, we will only vary the digits in the hundreds and tens places. Since the digit in the units place is the average of these two digits, these two digits must both be even or both be odd. We list the possible balanced numbers starting from the digit in the hundreds place.

| $1: \leadsto 111$ | 132 | 153 | 174 | 195 | $\leadsto$ | total of balanced numbers $1+4 \times 6=25$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $2: \leadsto 201$ | 222 | 243 | 264 | 285 | $\leadsto$ | $4+1+3 \times 6=23$ |
| $3: \leadsto$ | 333 | 354 | 375 | ; 396 | $\leadsto$ | $1+3 \times 6=19$ |
| $4: \leadsto$ | 402 | 444 | ; 465 | ; 486 | $\leadsto$ | $4+1+2 \times 6=17$ |
| $\leadsto$ |  | 555 | ; 576 | ; 597 | $\leadsto$ | $1+2 \times 6=13$ |
| $\leadsto$ |  | 603 | ; 666 | ; 687 | $\leadsto$ | $4+1+6=11$ |
|  |  |  | 777 | $; \quad 798$ | $\leadsto$ | $1+6=7$ |
|  |  |  | 804 | $; \quad 888$ | $\leadsto$ | $4+1=5$ |
|  |  |  |  | 999 | $\rightarrow$ |  |

Summing up, we have 121 balanced numbers of three digits.