8.6. On a certain segment, its endpoints and three internal points were marked. It turned out that all pairwise distances between the five marked points are different and are expressed in whole centimeters. What is the smallest possible length of the segment? Justify your answer.
Solution: There are 5 points, so there are 10 pairwise distances. If all of them are expressed as positive whole numbers of centimeters and are distinct, at least one of them is not less than 10. Therefore, the length of the segment is not less than 10. Suppose the length of the segment is exactly 10. Then the pairwise distances are all numbers from 1 to 10. Among them, there is a length of 9. This means that there must be a point that is 1 cm away from one of the segment's ends, let's say from the left end - see the top diagram, marked points are black. Then points that are 1 cm away from the right end and 2 cm away from the left (they are crossed out in the diagram) cannot be marked, otherwise there would be two segments of 1 cm. To achieve a length of 8 cm, a point must be marked 2 cm away from the right end. Now, points that are 1 or 2 cm away from the marked points (white points in the diagram) cannot be marked. Only two points remain, but each of them is the midpoint of a segment between some already marked points, so they cannot be marked either. Therefore, it is impossible to mark 5 points on a segment of length 10 in the required manner, and the length of the segment must be strictly greater than 10 cm.

This length is a whole number of centimeters, as it represents the distance between marked points - the ends of the segment. Therefore, it is not less than 11 cm. We will show the required arrangement of marked points on a segment of length 11. Let the marked points (from left to right) be points \(A, B, C, D\), and \(E\). Let \(AB = 1\), \(BC = 3\), \(CD = 5\), \(DE = 2\) - all distances in centimeters (see the bottom diagram). Then \(AC = 4\), \(AD = 9\), \(AE = 11\), \(BD = 8\), \(BE = 10\), \(CE = 7\) - all 10 distances are different.

Answer: 11 centimeters.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct justified answer | 7 points |
| Proven that marking points in the required manner on a segment of length 10 is impossible, but no example on a segment of length 11 cm | 5 points |
| Provided an example of the correct arrangement of points on a segment of length 11, but not justified that a length of 10 is impossible | 2 points |
| Correct answer without justification | 1 point |
| Incorrect examples of point placement OR correct placement of points on segments longer than 11 cm | 0 points |