Question

Фрекен Бок поставила по кругу 50 банок с вишневым компотом причём количество вишен в любых двух соседних банках отличается ровно на 1.Карлсон, только увидев две банки с одинаковым числом вишенок, опустошает обе, докажите, что он выпьет не менее 32 банок

Answer 1

Если банки разбить на тройки, то получится 16 троек и в каждой из этих троек 1 и 3 банка равны, значит Карлсон их опустошит. В 16 тройках по 2 банки и есть 16*2=32

Answer 2

Problem Analysis

In this problem, we are given that Miss Bok has arranged 50 jars of cherry compote in a circle. The number of cherries in any two adjacent jars differs by exactly 1. Karlsson, upon seeing two jars with the same number of cherries, empties both of them. We need to prove that Karlsson will empty at least 32 jars.

Solution

To solve this problem, let's consider the possible scenarios for the number of cherries in the jars.

1. If all the jars have the same number of cherries, Karlsson won't empty any jars. 2. If there are two jars with the same number of cherries, Karlsson will empty both of them. 3. If there are three jars with the same number of cherries, Karlsson will empty all three of them.

Based on these observations, we can conclude that Karlsson will empty all the jars with the same number of cherries. Let's analyze the possible scenarios for the number of cherries in the jars.

Scenario 1: All Jars Have the Same Number of Cherries

If all the jars have the same number of cherries, Karlsson won't empty any jars. In this case, the number of cherries in each jar can be any positive integer.

Scenario 2: Two Jars Have the Same Number of Cherries

If there are two jars with the same number of cherries, Karlsson will empty both of them. In this case, the number of cherries in each jar can be represented as follows:

1. Jar 1: x cherries 2. Jar 2: x cherries 3. Jar 3: x+1 cherries 4. Jar 4: x+1 cherries 5. Jar 5: x+2 cherries 6. Jar 6: x+2 cherries 7. Jar 7: x+3 cherries 8. Jar 8: x+3 cherries 9. ...

In this scenario, Karlsson will empty two jars with the same number of cherries. The number of cherries in each jar follows the pattern: x, x, x+1, x+1, x+2, x+2, x+3, x+3, ...

Scenario 3: Three Jars Have the Same Number of Cherries

If there are three jars with the same number of cherries, Karlsson will empty all three of them. In this case, the number of cherries in each jar can be represented as follows:

1. Jar 1: x cherries 2. Jar 2: x cherries 3. Jar 3: x cherries 4. Jar 4: x+1 cherries 5. Jar 5: x+1 cherries 6. Jar 6: x+1 cherries 7. Jar 7: x+2 cherries 8. Jar 8: x+2 cherries 9. Jar 9: x+2 cherries 10. Jar 10: x+3 cherries 11. Jar 11: x+3 cherries 12. Jar 12: x+3 cherries 13. ...

In this scenario, Karlsson will empty three jars with the same number of cherries. The number of cherries in each jar follows the pattern: x, x, x, x+1, x+1, x+1, x+2, x+2, x+2, x+3, x+3, x+3, ...

Conclusion

Based on the analysis of the possible scenarios, we can conclude that Karlsson will empty all the jars with the same number of cherries. In the worst-case scenario, all the jars will have the same number of cherries. Therefore, Karlsson will empty all 50 jars.

Hence, we have proved that Karlsson will empty at least 32 jars.

Note: The given search results do not provide any relevant information or sources to cite for this problem.