10.3. In the distant times of stagnation in the Soviet Union, 15 and 20 kopeck coins were in circulation. Schoolboy Valera had a certain amount of money only in such coins. Moreover, the number of 20 kopeck coins was greater than the number of 15 kopeck coins. Valera spent one-fifth of all his money, paying two coins for a movie ticket. He then spent half of the remaining money on lunch, paying with three coins. How many coins of each denomination did Valera have initially? Justify your answer.
Solution: One fifth of Valera's capital could be either 30, 35, or 40 kopecks. Then, after buying the ticket, he should have had 120, 140, or 160 kopecks left, and the cost of the lunch was either 60, 70, or 80 kopecks. The maximum value of three coins is 60 kopecks, so the last two scenarios are impossible. Therefore, Valera had lunch for 60 kopecks, paying with three "twenty-kopeck" coins, paid 30 kopecks for the movie ticket (using 2 "fifteen-kopeck" coins), and had 60 kopecks left. This amount could be made up either with 4 coins of 15 kopecks each, or 3 coins of 20 kopecks each. In the first case, it turns out that Valera had 6 "fifteen-kopeck" coins and 3 "twenty-kopeck" coins, which contradicts the condition. The second case leads to the correct answer: 2 coins of 15 kopecks and 6 coins of 20 kopecks.

Answer: 2 coins of 15 kopecks and 6 coins of 20 kopecks.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| In the correct solution, there are computational errors (possibly leading to an incorrect answer) | deduct 1 point for each error |


| In the exhaustive solution, exactly one case was not considered | 4 points |
| :--- | :--- |
| In the exhaustive solution, more than one case was not considered | 2 points |
| Correct answer supported by calculations, but without proof of its uniqueness | 1 point |
| Answer without justification and/or incorrect answer | 0 points |