4. If a two-digit number is decreased by 54, the result is a number written with the same digits but in reverse order. In the answer, specify the arithmetic mean of the resulting numbers.

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# Solution:

$\overline{x y}=10 x+y-$ the original two-digit number, then $\overline{y x}=10 y+x$ - the number written with the same digits but in reverse order. We get the equation $10 x+y=10 y+x+54$.

From the equation, it is clear that the two-digit number is greater than 54. Let's start the investigation with the tens digit equal to 6.

| $\boldsymbol{x}$ | equation | $\boldsymbol{y}$ | number |
| :---: | :---: | :---: | :---: |
| 6 | $60+y=10 y+6+54$ | $y=0$ | 60 does not fit the condition |
| 7 | $70+y=10 y+7+54$ | $y=1$ | 71 |
| 8 | $80+y=10 y+8+54$ | $y=2$ | 82 |
| 9 | $90+y=10 y+9+54$ | $y=3$ | 93 |

These could be the numbers 71, 82, 93. The arithmetic mean of the numbers is 82.

Answer: 82.