4. A three-digit number is called cool if one of its digits is half the product of the other two digits. A three-digit number is called supercool if such digits are in two or three of its positions. How many different supercool numbers exist? (Zero cannot be part of the representation of cool or supercool numbers.)
4. Answer: 25 numbers. Solution: let the record of a superclass number include digits a, b, c in some order. Then, two equalities are satisfied: $2 \mathrm{a}=\mathrm{bc} 2 \mathrm{~b}=\mathrm{ac}$. If the numbers a and b are different, then by swapping them, we will violate the existing equality. Therefore, $\mathrm{a}=\mathrm{b}$ and c $=2$. In this case, both equalities are automatically satisfied. This means that a superclass number consists of a two and two identical non-zero digits, and the two can be in any of the three available positions. This gives 24 options plus the number 222.

Criteria: correct solution - 7 points. All calculations are correct, but the equality of two digits is not explained - 3 points. The number 222 is forgotten - 4 points. Only the answer - 0 points. All numbers are listed or described in detail, but the absence of others is not proven - two points.