Problem 1. Given a natural number $n>1$, we perform the following operation: if $n$ is even, we divide it by two; if $n$ is odd, we add 5. If the number obtained after this operation is 1, we stop the process; otherwise, we apply the same operation again, and so on. Determine all values of $n$ for which this process is finite, that is, it reaches 1 at some point.
Solution. First, it is immediate to check that whenever we start with 2, 3, or 4, the process ends, and if we start with 5, we enter the loop $(5,10,5$, $10, \ldots)$ and never finish.

If the number we start with is greater than 5, in one or two steps we always move to a smaller number. To verify this, observe that if $n$ is even, it is evident; if it is odd, this follows from the inequality $\frac{n+5}{2}<n$, since $\frac{n+5}{2}$ is the number obtained after two iterations. Therefore, we always end up reaching a number less than or equal to 5, and we will only not terminate in those cases where we eventually reach 5 after any number of iterations.

Moreover, by successively applying the operations of adding five or dividing by two, we always maintain the property of being a multiple of 5. Thus, we simultaneously achieve the two results we want: (a) For those numbers that are not multiples of 5, by successively applying the operations, we always end up reaching 1, 2, 3, or 4, and therefore there is no cycle. (b) Similarly, any multiple of 5 will eventually reach 5, causing an infinite repetition that never reaches the number 1. Therefore, the process is finite if and only if $n$ is not a multiple of 5.