1. If a two-digit natural number is decreased by 54, the result is a two-digit number with the same digits but in reverse order. In the answer, specify the median of the sequence of all such numbers.

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# Solution.

Let $\overline{x y}=10 x+y$ be the original two-digit number, then $\overline{y x}=10 y+x$ is 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 median of the series is 82.

Answer: 82.