I have ten cards numbered from 1 to 10 divided to two stacks of 5 cards each. Let A be the product of the numbers on the 5 cards of the first stack and B be the product of the numbers on the 5 cards of the second stack. What is the minimum of the sum A+B

Given the problem with two stacks of cards, each containing 5 cards numbered from 1 to 10, we need to determine the minimum value of the sum \( A + B \), where:

- \( A \) is the product of the numbers on the first stack.
- \( B \) is the product of the numbers on the second stack.

### Key Insight:
Since the cards are divided into two stacks, the products \( A \) and \( B \) involve the multiplication of the numbers assigned to each stack. To minimize \( A + B \), we should distribute the numbers such that the two products \( A \) and \( B \) are as close to each other as possible.

### Approach:
1. The total product of all numbers from 1 to 10 is \( 10! \) (the factorial of 10).
   
   \[
   10! = 10 \times 9 \times 8 \times \ldots \times 2 \times 1 = 3,628,800
   \]

2. We want to split this product into two factors \( A \) and \( B \) such that the sum \( A + B \) is minimized. Ideally, \( A \) and \( B \) should be as close to \(\sqrt{10!}\) as possible.

3. Calculate \( \sqrt{10!} \):

   \[
   \sqrt{10!} \approx \sqrt{3,628,800} \approx 1904.68
   \]

4. We now need to find two subsets of the numbers \(\{1, 2, 3, \dots, 10\}\) such that their products \( A \) and \( B \) are as close as possible to 1904.68.

### Calculation:
Through trial or systematic check:

- Split one: \( A = 2 \times 3 \times 5 \times 7 \times 8 = 1680 \) and \( B = 1 \times 4 \times 6 \times 9 \times 10 = 2160 \)
- Sum: \( A + B = 1680 + 2160 = 3840 \)

This is a balanced split where \( A \) and \( B \) are relatively close.

### Conclusion:
The minimum possible sum \( A + B \) is \( \boxed{3840} \).

Would you like further details or have additional questions? Here are five related questions you might find interesting:

1. What are other potential splits of the numbers into two stacks?
2. How does the value of \( A + B \) change if we consider different product splits?
3. What mathematical strategies can be used to find optimal partitions?
4. How does this problem generalize to other numbers of cards?
5. What would happen if there were an odd number of cards?

**Tip:** In optimization problems, finding symmetry or balance between different parts often leads to an optimal solution.

Explore how to minimize the sum of products from two stacks of cards numbered 1 to 10. This problem involves calculating the factorial of 10 and distributing numbers to optimize two product sums. Suitable for high school students and math enthusiasts.

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