Problem 1166

To express the number 1988 as the sum of some positive integers such that the product of these integers is maximal, we can follow these steps:

1. Identify the optimal integers to use:
- If $a$ is a part of the sum and $a \geq 4$, we can replace $a$ with $2$ and $a - 2$. This is because $2(a - 2) \geq a$ for $a \geq 4$. This replacement increases the product while keeping the sum constant.
- Therefore, we should use only 2's and 3's in our sum.

2. Determine the optimal combination of 2's and 3's:
- Since $2^3 = 8$ and $3^2 = 9$, it is more beneficial to use as many 3's as possible because $3^2 > 2^3$.
- We need to express 1988 as a sum of 2's and 3's such that the number of 3's is maximized.

3. Calculate the number of 3's and 2's:
- We need to find the remainder when 1988 is divided by 3. This will help us determine how many 3's we can use.
- $1988 \div 3 = 662$ with a remainder of 2. This means we can use 662 threes and one two.

4. Form the sum and calculate the product:
- The sum is $3 \times 662 + 2 = 1986 + 2 = 1988$.
- The product is $3^{662} \times 2$.

Thus, the maximal product is achieved by using 662 threes and one two.

The final answer is $\boxed{ 2 \cdot 3^{662} }$.