For any sequence of real numbers A = {A_1, A_2, A_3, ... }, define *A to be the sequence {A_2 - A_1, A_3 - A_2, A_4 - A_3,..}. Suppose that all of the terms of the sequence *(*A) are 1, and that A_19 = A_92 = 0. Find A_1.

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To solve this problem, we need to use the given information and the definition of the sequence *A to find A_1. Let's break down the problem step by step:

Step 1: Using the definition of the sequence *A, we know that the terms of *(*A) are all 1. This means that:
(*A)_1 = (*A)_2 = (*A)_3 = ... = 1.

Step 2: We need to find a relation between the terms of A and the terms of *(*A). Let's consider the definition of *(*A):
(*A) = {A_2 - A_1, A_3 - A_2, A_4 - A_3, ... }.

From this definition, we can observe that the difference between consecutive terms in *(*A) is the same as the difference between consecutive terms in A. So, we can write:
(*A)_1 = A_2 - A_1,
(*A)_2 = A_3 - A_2,
(*A)_3 = A_4 - A_3,
...

Step 3: To find A_1, we can start by finding the value of (*A)_1:
(*A)_1 = 1, as given in the problem statement.

But we also know that (*A)_1 = A_2 - A_1, as we derived in Step 2. Therefore, we can substitute (*A)_1 with its equivalent expression:
A_2 - A_1 = 1.

Step 4: We are given that A_19 = A_92 = 0. Using this information, we can rewrite the equation obtained in Step 3 as:
A_2 - 0 = 1,
A_2 = 1.

Step 5: Now that we know A_2, we can use the definition of *(*A) to find A_1:
(*A)_1 = A_2 - A_1.
Substituting the values (*A)_1 = 1 and A_2 = 1:
1 = 1 - A_1.

Step 6: Solve the equation from Step 5 for A_1:
1 = 1 - A_1,
A_1 = 1 - 1,
A_1 = 0.

Therefore, the value of A_1 is 0.

By following these steps, we have determined that A_1 = 0.