Problem 9.8. Young entomologist Dima is observing two grasshoppers. He noticed that when a grasshopper starts jumping, it jumps 1 cm, then after a second, 2 cm, then another second, 3 cm, and so on.

Initially, both grasshoppers were in the same place. One of them started jumping, and after a few seconds, the second one started jumping after the first (the grasshoppers jump along a straight line in the same direction). At some point, Dima wrote in his notebook that the distance between the grasshoppers was 9 cm. A few seconds later, he wrote that the distance between the grasshoppers had become 39 cm. How many seconds passed between the recordings? (List all possible options.)
Answer: $10,15,30$.

Solution. Note that the grasshoppers are one or several consecutive jumps apart from each other. Thus, the distance between them is the sum of $m$ consecutive natural numbers, where $m$ is the number of seconds by which the second grasshopper started later.

On the other hand, four consecutive jumps already sum up to no less than $1+2+3+4=10$, which means the number of jumps by which the grasshoppers are apart cannot exceed 3. Therefore, the possible values for $m$ are 1, 2, and 3. It is easy to see that all these cases are possible: a distance of 9 can be achieved with three jumps $(2+3+4)$, two jumps $(4+5)$, or one jump.

If there are $m$ jumps between the grasshoppers, then each second the distance between them will increase by $m$, since each of the $m$ terms corresponding to individual jumps increases by 1. Thus, the distance can increase by 30 in $30/m$ seconds. Substituting $m=1,2,3$, we get the answers.