Introduction:
To find the maximum value of A1 A2 A3 ... A20, where A1, A2, A3, ..., A20 are arithmetic means between 13 and 67, we need to determine the common difference (d) of the arithmetic progression formed by these terms.

Given Information:
- A1, A2, A3, ..., A20 are arithmetic means between 13 and 67.

Approach:
Step 1: Find the common difference (d):
We know that the formula for the nth term of an arithmetic progression is given by An = A + (n - 1)d, where A is the first term, n is the number of terms, and d is the common difference.

In this case, since A1, A2, A3, ..., A20 are arithmetic means, there are a total of 20 terms. Let's assume the first term (A) is A1 and the common difference (d) is d.

Using the formula, we can write:
A20 = A1 + (20 - 1)d
67 = 13 + 19d
54 = 19d
d = 54/19 = 2.8421 (approx.)

Therefore, the common difference (d) is approximately 2.8421.

Step 2: Find the maximum value:
To find the maximum value of A1 A2 A3 ... A20, we need to determine the value of A1.

Using the formula for the nth term, we can write:
A1 = A + (1 - 1)d
A1 = A

Since A1 is the first term, it will be equal to the given minimum value, which is 13.

Now, to find the maximum value, we can substitute the values of A1 and d into the formula for the nth term:
A20 = A1 + (20 - 1)d
A20 = 13 + 19(2.8421)
A20 ≈ 13 + 54.0000
A20 ≈ 67 (approx.)

Therefore, the maximum value of A1 A2 A3 ... A20 is approximately 67.

Conclusion:
The maximum value of A1 A2 A3 ... A20, where A1, A2, A3, ..., A20 are arithmetic means between 13 and 67, is approximately 67.

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