## Task 6 - 070836

A two-digit natural number is sought with the following properties:

1) The difference between its digits is three.
2) If its digits are swapped, the new number is nine less than twice the original number.
(I) Since the number is supposed to be two-digit, it must be greater than 9. This implies that its double, reduced by 9, must be greater than the number itself.

Due to the second condition, this means that by reversing the digits of the number, a larger number should result. Therefore, its first digit must be smaller than its second.

(II) Furthermore, the double reduced by 9 should again be two-digit, i.e., at most 99.

This implies that the number can be at most 54. Given the first condition, the following possibilities remain: 14, 25, 36, and 47.

Of these, only the number 36 satisfies all the conditions of the problem.

Solutions of the III. Round 1967 taken from [5]

### 5.10 VIII. Olympiad 1968

### 5.10.1 I. Round 1968, Class 8