## Task 2 - 201212

Four people $A, B, C, D$ make two statements each about a non-negative integer $x$ written in decimal notation. It is known that

(1) exactly one of $A, B, C$ makes two false statements, while each of the other two makes exactly one false statement,

(2) $D$ makes two true statements.

The statements made by $A, B, C, D$ are:

(A1) The last digit of the decimal representation of $x$ is even.

(A2) $x$ is a perfect square.

(B1) The digit 9 is present at least once in the decimal representation of $x$.

(B2) $x$ is a four-digit number.

(C1) $x$ is divisible by 10.

(C2) $x$ leaves a remainder of 1 when divided by 3.

(D1) In the decimal representation of $x$, if $x$ consists of more than one digit, each digit from left to right is 1 less than the digit immediately to its right.

(D2) The number of even digits in the decimal representation of $x$ is not greater than 2.

Determine all numbers $x$ that satisfy this system of conditions!
If for a number $x$ the statements (D1) and (D2) are true, it follows that the digits of $x$ alternate between even and odd.

Furthermore, it follows that at most two even digits and - alternating with them - at most three odd digits occur. Thus, the number $x$ is at most five digits long, and if it is five digits long, it begins with an odd digit. Therefore, (D1) and (D2) can only be true for the numbers $x$ in the following table.

Conversely, (D1) and (D2) are confirmed for all these numbers. They are thus exactly the numbers that satisfy condition (2). For each of them, the table indicates whether the statements (A1) to (C2) are true or false. This yields the subsequently mentioned counts of false statements from A, B, and C.

| $x$ | (A1) | (A2) | (B1) | (B2) | (C1) | (C2) | false A | statements B | of C |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 0 | $\mathrm{~W}$ | $\mathrm{~W}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~W}$ | $\mathrm{~F}$ | 0 | 2 | 1 |
| 1 | $\mathrm{~F}$ | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | W | 1 | 2 | 1 |
| 2 | W | F | $\mathrm{F}$ | $\mathrm{F}$ | F | $\mathrm{F}$ | 1 | 2 | 2 |
| 3 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 2 | 2 |
| 4 | W | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | W | 0 | 2 | 1 |
| 5 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 2 | 2 |
| 6 | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 1 | 2 | 2 |
| 7 | F | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | W | 2 | 2 | 1 |
| 8 | W | $\mathrm{F}$ | F | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 1 | 2 | 2 |
| 9 | $\mathrm{~F}$ | W | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 1 | 1 | 2 |
| 12 | W | F | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 1 | 2 | 2 |
| 23 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 2 | 2 |
| 34 | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{W}$ | 1 | 2 | 1 |
| 45 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 2 | 2 |
| 56 | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 1 | 2 | 2 |
| 67 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | W | 2 | 2 | 1 |
| 78 | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 1 | 2 | 2 |
| 89 | $\mathrm{~F}$ | $\mathrm{~F}$ | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 2 | 2 | 2 |
| 123 | $\mathrm{~F}$ | $\mathrm{~F}$ | F | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 2 | 2 | 2 |
| 234 | W | $\mathrm{F}$ | $\mathrm{F}$ | F | $\mathrm{F}$ | $\mathrm{F}$ | 1 | 2 | 2 |
| 345 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 2 | 2 |
| 456 | W | F | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | F | 1 | 2 | 2 |
| 567 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 2 | 2 |
| 678 | W | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 1 | 2 | 2 |
| 789 | $\mathrm{~F}$ | $\mathrm{~F}$ | w | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | 2 | 1 | 2 |

| $x$ |  |  |  |  |  |  | false | statements | of |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $x$ | $(\mathrm{~A} 1)$ | $(\mathrm{A} 2)$ | $(\mathrm{B} 1)$ | (B2) | (C1) | (C2) | A | B | C |
| 1234 | $\mathrm{~W}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~W}$ | $\mathrm{~F}$ | $\mathrm{~W}$ | 1 | 1 | 1 |
| 2345 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~W}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 1 | 2 |
| $\mathbf{3 4 5 6}$ | $\mathrm{W}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{W}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ |
| $\mathbf{4 5 6 7}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{W}$ | $\mathrm{F}$ | $\mathrm{W}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ |
| $\mathbf{5 6 7 8}$ | $\mathrm{W}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathrm{W}$ | $\mathrm{F}$ | $\mathrm{F}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ |
| 6789 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~W}$ | $\mathrm{~W}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 0 | 2 |
| 12345 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 2 | 2 |
| 34567 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~W}$ | 2 | 2 | 1 |
| 56789 | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~W}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | $\mathrm{~F}$ | 2 | 1 | 2 |

From this, it is evident that among all numbers that satisfy (2), exactly the numbers 1, 9, 34, 3456, 4567, 5678 also satisfy condition (1). Thus, these are exactly the numbers sought.

## Adopted from $[5]$