Vanya took three Unified State Exams (USE). He scored 5 points less in Russian than in Physics, and in Physics, he scored 9 points less than in Mathematics. The golden fish that appeared in Vanya's dream promised to fulfill any number of wishes of the following types:
- add one point to each exam; - decrease the score by 3 for one exam (of Vanya's choice), and increase the score by 1 for each of the other two exams.
The fish will fulfill a wish if none of the results exceed 100 points. Could Vanya, in his dream, score 100 points on more than one exam?
This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.
Solution: For Vanya to score more than 100 points on more than one exam, the difference in points between these two exams must become zero. However, the difference in points between any two exams either remains unchanged or changes by 4. And initially, the difference in points does not divide by 4 for any two exam results.
Source: NuminaMath-1.5,
licensed Apache-2.0.
Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.