## Task 4 - 130624

Werner writes $50^{*} 0 * 05$ on the board and wants to insert one of the digits $0,1,2$, $3,4,5,6,7,8,9$ for each of the symbols * so that a number divisible by 9 is formed.

Give all possible ways of such an insertion (i.e., all such obtainable numbers that are divisible by 9)!
The sum of the digits to be entered for the characters * is at least 0 and at most 18.

The cross sum of the number without these digits is 10. Therefore, the cross sum of the desired number is at least 10 and at most 28.

On the other hand, a number is divisible by 9 if and only if its cross sum is divisible by 9. Therefore, an entry meets the conditions of the problem if and only if the cross sum of the resulting number is either 18 or 27, which is the case if and only if the sum of the two digits to be entered is 8 or 17.

Consequently, there are exactly the following numbers that meet the conditions of the problem and are divisible by 9:

$5000805,5010705,5020605,5030505,5040405,5050305$,

$5060205,5070105,5080005,5080905,5090805$.

Solutions of the II. Round 1973 taken from [5]

### 3.15 XIV. Olympiad 1974

### 3.15.1 I. Round 1974, Class 6