4. A sequence of 300 natural numbers is written in a row. Each number, starting from the third, is equal to the product of the two preceding numbers. How many perfect squares can there be among these numbers? (Provide all answers and prove that there are no others.)
Answer: 0, 100 or 300. If after number a, number $b$ is written, then the numbers аb and $а b^{2}$ follow them. The number $a b^{2}$ is a perfect square if and only if a is a perfect square. Among the first three numbers, squares can be all three, one, or none (examples of all three situations are trivial). Hence, the answer follows immediately.