## Task 4 - 110524

Determine all two-digit natural numbers $z$ that each satisfy the following conditions simultaneously:

(a) The number $z$ is not divisible by 10.

(b) If you subtract the units digit of the number from its tens digit, you get 4.

(c) If you swap the digits of $z$ with each other, you get a new two-digit number $z_{1}$, whose triple is smaller than $z$.
Condition (1) is satisfied if and only if the unit digit of the sought numbers is not 0. Therefore, conditions (1) and (2) are exactly satisfied by the numbers 51, 62, 73, 84, 95.

If one swaps their digits, one obtains the numbers 15, 26, 37, 48, 59 in sequence.

The triple of these numbers is 45, 78, 111, 144, 177 in sequence. Since $45 < 51$, $78 < 62$, $111 > 73$, $144 > 84$, $177 > 95$, only the number 51 satisfies all conditions of the problem.

Solution taken from [5]

### 2.13 XII. Olympiad 1972

### 2.13.1 First Round 1972, Class 5