Subject 4, author ***

a) On a circle, natural numbers from 1 to 20 are written in order, in the clockwise direction. By a step, we mean swapping the positions of any two of these 20 numbers. What is the minimum number of steps required to rearrange these numbers in reverse order, in the counterclockwise direction?

b) What is the answer to the question in a) if 21 numbers are written on the circle in the clockwise direction from 1 to 21?

| Details of Solution | Associated Scoring |
| :--- | :---: |
| A) We leave the numbers 1 and 11 in place and swap the following pairs: $(2,20),(3,19),(4,18)$,  $(5,17),(6,16),(7,15),(8,14),(9,13),(10,12)$ | 2 |
| The minimum number of swaps is 9, because with 8 swaps, 4 numbers remain in increasing order in the clockwise direction | 2 |
| B) We fix one number, for example, 1. We swap the following pairs: $(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(11,12)$ | 2 |
| The minimum number of swaps is 10, because with 9 swaps, 3 numbers remain in increasing order in the clockwise direction. | 1 |
4. a) On a circle, the natural numbers from 1 to 20 are written in order in a clockwise direction. By a step, we mean to swap the positions of any two of the 20 numbers written on the circle. What is the minimum number of steps required to rearrange these numbers successively in the counterclockwise direction?

b) What is the answer to the question in a) if 21 numbers are written on the circle in a clockwise direction from 1 to 21?

## MATHEMATICS OLYMPIAD  SECTOR STAGE, 21.02.2016  5th GRADE  SOLUTIONS AND ORIENTATIVE SCORING GUIDELINES

Note: Each problem is scored from 0 to 7 points. Only integer scores are given. Any other solution is assimilated according to the scoring guidelines.