**Introduction**

In this problem, we have two persons, A and B, who are running between two points, P and Q, infinitely. A starts from point P, and B starts from point Q. We are given that they meet for the first time at a distance of 0.6D from point P, where D is the distance between points P and Q. We need to determine at how many distinct points they will meet until their 10th meeting.

**Understanding the Problem**

To solve this problem, we need to understand the concept of relative speed. The relative speed between A and B is the difference in their speeds. Let's assume that the speed of A is x units per time and the speed of B is y units per time. The relative speed between them will be (x - y) units per time.

**Determining the Distance Covered**

Since A and B meet for the first time at a distance of 0.6D from point P, we can say that A has covered 0.6D distance and B has covered (D - 0.6D) = 0.4D distance. Therefore, the ratio of the distance covered by A to the distance covered by B is 0.6D:0.4D = 3:2.

**Ratio of Speeds**

The ratio of the speeds of A and B is the inverse of the ratio of the distances covered by them. Therefore, the ratio of the speeds of A and B is 2:3.

**Finding the Relative Speed**

Since we know the ratio of the speeds of A and B, we can assume any values for their speeds as long as the ratio remains the same. Let's assume that the speed of A is 2x units per time and the speed of B is 3x units per time. Therefore, the relative speed between A and B will be (2x - 3x) = -x units per time.

**Determining the Meeting Points**

When two objects are moving in opposite directions with a relative speed, they meet at regular intervals. In this case, A and B are moving towards each other with a relative speed of -x units per time. Therefore, they will meet at regular intervals of D/|(-x)| units of time.

Since they meet for the first time at a distance of 0.6D from point P, the time taken for the first meeting will be (0.6D)/(x) units of time. We can generalize this as follows:

**The time taken for the nth meeting is given by (0.6D)/(x) * n units of time.**

**Calculating the Number of Distinct Meeting Points**

To determine the number of distinct meeting points till the 10th meeting, we need to calculate the number of distinct values of n for which the time taken for the nth meeting is less than or equal to 10 units of time.

Let's substitute the given values into the equation and solve for n:

(0.6D)/(x) * n ≤ 10

n ≤ (10 * x)/(0.6D)

Since n represents the number of meetings, it should be an integer. Therefore, the largest integer less than or equal to (10 * x)/(0.6D) will give us the number of distinct meeting points till the 10th meeting.

**Conclusion**

In conclusion, the number of distinct

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