Problem 2. Vasya solved problems for 10 days - at least one problem each day. Each day (except the first), if the weather was cloudy, he solved one more problem than the previous day, and if it was sunny, one less problem. In the first 9 days, Vasya solved 13 problems. What was the weather like on the tenth day? [5 points]

(B. Frenkin)
Answer: Cloudy.

Solution. Consider any two consecutive days. On each day, at least one problem is solved, but it cannot be exactly one on both days, so at least three problems are solved over these two days. Thus, over the first 8 days, Vasya solved at least \(4 \cdot 3 = 12\) problems. If he solved at least two problems on the ninth day, the total number of problems solved in 9 days would exceed 13. Therefore, exactly one problem was solved on the ninth day. On the tenth day, the weather was cloudy (and Vasya solved two problems), otherwise he would have solved 0 problems on that day, which contradicts the condition.

An example of how this could have happened: Vasya solved \(1, 2, 1, 2, 1, 2, 1, 2, 1\), 2 problems over these 10 days. It is not difficult to prove that this example is unique. Indeed, the proof that Vasya solved exactly one problem on the ninth day can be applied to any odd day.

Comment. It might seem that the answer to the question could be given without considering either the number of days or the total number of problems solved. Indeed, cloudy weather on the last day does not contradict anything, while sunny weather can, and precisely in the case when exactly one problem was solved on the penultimate day. Since we are asked to determine the weather on the last day, the answer is "cloudy." Nevertheless, such reasoning cannot be considered a solution, as it is based on the implicit assumption that the problem has a unique answer.