We are asked to evaluate the expression $A = 2025^3 - 2024 \cdot 2025^2 - 2024^2 \cdot 2025 + 2024^3$.

1. Problem Description

We are asked to evaluate the expression $A = 2025^3 - 2024 \cdot 2025^2 - 2024^2 \cdot 2025 + 2024^3$.

2. Solution Steps

Let $a = 2025$ and $b = 2024$. Then the expression becomes:

$A = a^3 - b a^2 - b^2 a + b^3$.

We can rewrite this as:

$A = a^3 + b^3 - b a^2 - b^2 a$

$A = (a^3 + b^3) - (b a^2 + b^2 a)$

We can factor each term:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

$b a^2 + b^2 a = ab(a + b)$

So, $A = (a+b)(a^2 - ab + b^2) - ab(a+b)$

We can factor out $(a+b)$:

$A = (a+b)(a^2 - ab + b^2 - ab)$

$A = (a+b)(a^2 - 2ab + b^2)$

The term $(a^2 - 2ab + b^2)$ is a perfect square:

$a^2 - 2ab + b^2 = (a-b)^2$

So, $A = (a+b)(a-b)^2$

Now, we substitute $a = 2025$ and $b = 2024$:

$A = (2025 + 2024)(2025 - 2024)^2$

$A = (4049)(1)^2$

$A = 4049 \cdot 1$

$A = 4049$

3. Final Answer

The final answer is 4049.