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задача: Кенгуру 9-10 класс - 10 чисел по кругу

Задачу прислал: Админ


Сложность: средняяЧисла 1,2,3..10 выписаны по кругу в произвольном порядке. Складывая каждое из этих чисел с двумя его "соседями", мы получим 10 сумм. Пусть А - это наименьшая из этих сумм. Найдите наибольшее возможное значение А.



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Решение задачи





Ваши ответы на задачу


ответов: 10

venik 2013-07-31 21:43:51 пишет:
27

евгений 2013-04-05 11:59:41 пишет:
[5, 7, 3, 8, 6, 1, 10, 4, 2, 9]
[5, 7, 8, 1, 6, 9, 2, 4, 10, 3]
[5, 7, 8, 1, 6, 9, 4, 2, 10, 3]
[5, 7, 8, 1, 6, 10, 2, 4, 9, 3]
[5, 7, 9, 2, 4, 10, 1, 6, 8, 3]
[5, 8, 6, 1, 10, 4, 2, 9, 7, 3]
[5, 9, 2, 4, 10, 1, 6, 8, 3, 7]
[5, 9, 2, 4, 10, 1, 6, 8, 7, 3]
[5, 9, 2, 4, 10, 1, 8, 6, 3, 7]
[5, 9, 2, 4, 10, 1, 8, 6, 7, 3]
[5, 9, 2, 4, 10, 6, 1, 8, 7, 3]
[5, 9, 2, 6, 7, 3, 8, 4, 10, 1]
[5, 9, 2, 6, 7, 8, 3, 4, 10, 1]
[5, 9, 2, 7, 6, 3, 8, 4, 10, 1]
[5, 9, 2, 7, 6, 8, 3, 4, 10, 1]
[5, 9, 3, 4, 8, 6, 2, 7, 10, 1]
[5, 9, 3, 4, 8, 7, 2, 6, 10, 1]
[5, 9, 4, 2, 10, 6, 1, 8, 7, 3]
[5, 9, 4, 3, 8, 6, 2, 7, 10, 1]
[5, 9, 4, 3, 8, 7, 2, 6, 10, 1]
[5, 9, 6, 2, 7, 8, 3, 4, 10, 1]
[5, 9, 7, 2, 6, 8, 3, 4, 10, 1]
[5, 10, 2, 4, 9, 6, 1, 8, 7, 3]
[5, 10, 3, 4, 8, 6, 2, 7, 9, 1]
[5, 10, 3, 4, 8, 7, 2, 6, 9, 1]
[5, 10, 4, 2, 9, 6, 1, 8, 7, 3]
[5, 10, 4, 3, 8, 6, 2, 7, 9, 1]
[5, 10, 4, 3, 8, 7, 2, 6, 9, 1]
[6, 1, 8, 7, 3, 5, 9, 2, 4, 10]
[6, 1, 8, 7, 3, 5, 9, 4, 2, 10]
[6, 1, 8, 7, 3, 5, 10, 2, 4, 9]
[6, 1, 8, 7, 3, 5, 10, 4, 2, 9]
[6, 1, 8, 7, 5, 3, 9, 4, 2, 10]
[6, 1, 8, 7, 5, 3, 10, 2, 4, 9]
[6, 1, 8, 7, 5, 3, 10, 4, 2, 9]
[6, 1, 10, 4, 2, 9, 5, 3, 7, 8]
[6, 1, 10, 4, 2, 9, 5, 7, 3, 8]
[6, 1, 10, 4, 2, 9, 7, 3, 5, 8]
[6, 1, 10, 4, 2, 9, 7, 5, 3, 8]
[6, 2, 7, 8, 3, 4, 9, 5, 1, 10]
[6, 2, 7, 8, 3, 4, 10, 1, 5, 9]
[6, 2, 7, 8, 3, 4, 10, 5, 1, 9]
[6, 2, 7, 8, 4, 3, 9, 5, 1, 10]
[6, 2, 7, 8, 4, 3, 10, 5, 1, 9]
[6, 2, 7, 9, 1, 5, 10, 3, 4, 8]
[6, 2, 7, 9, 1, 5, 10, 4, 3, 8]
[6, 2, 7, 9, 5, 1, 10, 4, 3, 8]
[6, 2, 7, 10, 1, 5, 9, 3, 4, 8]
[6, 2, 7, 10, 1, 5, 9, 4, 3, 8]
[6, 2, 9, 5, 1, 10, 4, 3, 8, 7]
[6, 2, 9, 5, 1, 10, 4, 8, 3, 7]
[6, 3, 7, 5, 9, 2, 4, 10, 1, 8]
[6, 3, 8, 4, 10, 1, 5, 9, 2, 7]
[6, 7, 2, 9, 5, 1, 10, 4, 3, 8]
[6, 7, 2, 9, 5, 1, 10, 4, 8, 3]
[6, 7, 3, 5, 9, 2, 4, 10, 1, 8]
[6, 7, 3, 8, 4, 10, 1, 5, 9, 2]
[6, 7, 8, 3, 4, 10, 1, 5, 9, 2]
[6, 8, 1, 10, 4, 2, 9, 5, 3, 7]
[6, 8, 1, 10, 4, 2, 9, 5, 7, 3]
[6, 8, 3, 4, 9, 5, 1, 10, 7, 2]
[6, 8, 3, 4, 10, 1, 5, 9, 2, 7]
[6, 8, 3, 4, 10, 1, 5, 9, 7, 2]
[6, 8, 3, 4, 10, 5, 1, 9, 7, 2]
[6, 8, 3, 5, 7, 9, 2, 4, 10, 1]
[6, 8, 3, 7, 5, 9, 2, 4, 10, 1]
[6, 8, 4, 3, 9, 5, 1, 10, 7, 2]
[6, 8, 4, 3, 10, 5, 1, 9, 7, 2]
[6, 8, 5, 3, 7, 9, 2, 4, 10, 1]
[6, 8, 7, 3, 5, 9, 2, 4, 10, 1]
[6, 9, 1, 5, 10, 3, 4, 8, 7, 2]
[6, 9, 1, 5, 10, 4, 3, 8, 7, 2]
[6, 9, 2, 4, 10, 3, 5, 7, 8, 1]
[6, 9, 2, 4, 10, 5, 3, 7, 8, 1]
[6, 9, 4, 2, 10, 3, 5, 7, 8, 1]
[6, 9, 4, 2, 10, 5, 3, 7, 8, 1]
[6, 9, 5, 1, 10, 4, 3, 8, 7, 2]
[6, 10, 1, 5, 9, 3, 4, 8, 7, 2]
[6, 10, 1, 5, 9, 4, 3, 8, 7, 2]
[6, 10, 2, 4, 9, 3, 5, 7, 8, 1]
[6, 10, 2, 4, 9, 5, 3, 7, 8, 1]
[6, 10, 4, 2, 9, 5, 3, 7, 8, 1]
[7, 2, 6, 8, 3, 4, 9, 5, 1, 10]
[7, 2, 6, 8, 3, 4, 10, 1, 5, 9]
[7, 2, 6, 8, 3, 4, 10, 5, 1, 9]
[7, 2, 6, 8, 4, 3, 9, 5, 1, 10]
[7, 2, 6, 8, 4, 3, 10, 5, 1, 9]
[7, 2, 6, 9, 1, 5, 10, 3, 4, 8]
[7, 2, 6, 9, 1, 5, 10, 4, 3, 8]
[7, 2, 6, 9, 5, 1, 10, 4, 3, 8]
[7, 2, 6, 10, 1, 5, 9, 3, 4, 8]
[7, 2, 6, 10, 1, 5, 9, 4, 3, 8]
[7, 2, 9, 5, 1, 10, 4, 3, 8, 6]
[7, 2, 9, 5, 1, 10, 4, 8, 3, 6]
[7, 3, 5, 8, 6, 1, 10, 4, 2, 9]
[7, 3, 5, 9, 2, 4, 10, 1, 6, 8]
[7, 3, 5, 9, 2, 4, 10, 1, 8, 6]
[7, 3, 5, 9, 2, 4, 10, 6, 1, 8]
[7, 3, 5, 9, 4, 2, 10, 6, 1, 8]
[7, 3, 5, 10, 2, 4, 9, 6, 1, 8]
[7, 3, 5, 10, 4, 2, 9, 6, 1, 8]
[7, 3, 6, 8, 1, 10, 4, 2, 9, 5]
[7, 3, 8, 4, 10, 1, 5, 9, 2, 6]
[7, 3, 8, 6, 1, 10, 4, 2, 9, 5]
[7, 5, 3, 8, 6, 1, 10, 4, 2, 9]
[7, 5, 3, 9, 4, 2, 10, 6, 1, 8]
[7, 5, 3, 10, 2, 4, 9, 6, 1, 8]
[7, 5, 3, 10, 4, 2, 9, 6, 1, 8]
[7, 5, 9, 2, 4, 10, 1, 6, 8, 3]
[7, 5, 9, 2, 4, 10, 1, 8, 6, 3]
[7, 6, 2, 9, 5, 1, 10, 4, 3, 8]
[7, 6, 2, 9, 5, 1, 10, 4, 8, 3]
[7, 6, 3, 8, 4, 10, 1, 5, 9, 2]
[7, 6, 8, 1, 10, 4, 2, 9, 5, 3]
[7, 6, 8, 3, 4, 10, 1, 5, 9, 2]
[7, 8, 1, 6, 9, 2, 4, 10, 3, 5]
[7, 8, 1, 6, 9, 2, 4, 10, 5, 3]
[7, 8, 1, 6, 9, 4, 2, 10, 3, 5]
[7, 8, 1, 6, 9, 4, 2, 10, 5, 3]
[7, 8, 1, 6, 10, 2, 4, 9, 3, 5]
[7, 8, 1, 6, 10, 2, 4, 9, 5, 3]
[7, 8, 1, 6, 10, 4, 2, 9, 5, 3]
[7, 8, 3, 4, 9, 5, 1, 10, 6, 2]
[7, 8, 3, 4, 10, 1, 5, 9, 2, 6]
[7, 8, 3, 4, 10, 1, 5, 9, 6, 2]
[7, 8, 3, 4, 10, 5, 1, 9, 6, 2]
[7, 8, 4, 3, 9, 5, 1, 10, 6, 2]
[7, 8, 4, 3, 10, 5, 1, 9, 6, 2]
[7, 8, 6, 1, 10, 4, 2, 9, 5, 3]
[7, 9, 1, 5, 10, 3, 4, 8, 6, 2]
[7, 9, 1, 5, 10, 4, 3, 8, 6, 2]
[7, 9, 2, 4, 10, 1, 6, 8, 3, 5]
[7, 9, 2, 4, 10, 1, 6, 8, 5, 3]
[7, 9, 5, 1, 10, 4, 3, 8, 6, 2]
[7, 10, 1, 5, 9, 3, 4, 8, 6, 2]
[7, 10, 1, 5, 9, 4, 3, 8, 6, 2]
[8, 1, 6, 9, 2, 4, 10, 3, 5, 7]
[8, 1, 6, 9, 2, 4, 10, 5, 3, 7]
[8, 1, 6, 9, 4, 2, 10, 3, 5, 7]
[8, 1, 6, 9, 4, 2, 10, 5, 3, 7]
[8, 1, 6, 10, 2, 4, 9, 3, 5, 7]
[8, 1, 6, 10, 2, 4, 9, 5, 3, 7]
[8, 1, 6, 10, 4, 2, 9, 5, 3, 7]
[8, 1, 10, 4, 2, 9, 5, 3, 7, 6]
[8, 1, 10, 4, 2, 9, 5, 7, 3, 6]
[8, 3, 4, 9, 5, 1, 10, 6, 2, 7]
[8, 3, 4, 9, 5, 1, 10, 7, 2, 6]
[8, 3, 4, 10, 1, 5, 9, 2, 6, 7]
[8, 3, 4, 10, 1, 5, 9, 2, 7, 6]
[8, 3, 4, 10, 1, 5, 9, 6, 2, 7]
[8, 3, 4, 10, 1, 5, 9, 7, 2, 6]
[8, 3, 4, 10, 5, 1, 9, 6, 2, 7]
[8, 3, 4, 10, 5, 1, 9, 7, 2, 6]
[8, 3, 5, 7, 9, 2, 4, 10, 1, 6]
[8, 3, 6, 7, 2, 9, 5, 1, 10, 4]
[8, 3, 7, 5, 9, 2, 4, 10, 1, 6]
[8, 3, 7, 6, 2, 9, 5, 1, 10, 4]
[8, 4, 3, 9, 5, 1, 10, 6, 2, 7]
[8, 4, 3, 9, 5, 1, 10, 7, 2, 6]
[8, 4, 3, 10, 5, 1, 9, 6, 2, 7]
[8, 4, 3, 10, 5, 1, 9, 7, 2, 6]
[8, 4, 10, 1, 5, 9, 2, 6, 7, 3]
[8, 4, 10, 1, 5, 9, 2, 7, 6, 3]
[8, 5, 3, 7, 9, 2, 4, 10, 1, 6]
[8, 6, 1, 10, 4, 2, 9, 5, 3, 7]
[8, 6, 1, 10, 4, 2, 9, 5, 7, 3]
[8, 6, 1, 10, 4, 2, 9, 7, 3, 5]
[8, 6, 1, 10, 4, 2, 9, 7, 5, 3]
[8, 6, 2, 7, 9, 1, 5, 10, 3, 4]
[8, 6, 2, 7, 9, 1, 5, 10, 4, 3]
[8, 6, 2, 7, 9, 5, 1, 10, 4, 3]
[8, 6, 2, 7, 10, 1, 5, 9, 3, 4]
[8, 6, 2, 7, 10, 1, 5, 9, 4, 3]
[8, 6, 3, 7, 5, 9, 2, 4, 10, 1]
[8, 6, 7, 2, 9, 5, 1, 10, 4, 3]
[8, 6, 7, 3, 5, 9, 2, 4, 10, 1]
[8, 7, 2, 6, 9, 1, 5, 10, 3, 4]
[8, 7, 2, 6, 9, 1, 5, 10, 4, 3]
[8, 7, 2, 6, 9, 5, 1, 10, 4, 3]
[8, 7, 2, 6, 10, 1, 5, 9, 3, 4]
[8, 7, 2, 6, 10, 1, 5, 9, 4, 3]
[8, 7, 3, 5, 9, 2, 4, 10, 1, 6]
[8, 7, 3, 5, 9, 2, 4, 10, 6, 1]
[8, 7, 3, 5, 9, 4, 2, 10, 6, 1]
[8, 7, 3, 5, 10, 2, 4, 9, 6, 1]
[8, 7, 3, 5, 10, 4, 2, 9, 6, 1]
[8, 7, 5, 3, 9, 4, 2, 10, 6, 1]
[8, 7, 5, 3, 10, 2, 4, 9, 6, 1]
[8, 7, 5, 3, 10, 4, 2, 9, 6, 1]
[8, 7, 6, 2, 9, 5, 1, 10, 4, 3]
[9, 1, 5, 10, 3, 4, 8, 6, 2, 7]
[9, 1, 5, 10, 3, 4, 8, 7, 2, 6]
[9, 1, 5, 10, 4, 3, 8, 6, 2, 7]
[9, 1, 5, 10, 4, 3, 8, 7, 2, 6]
[9, 2, 4, 10, 1, 6, 8, 3, 5, 7]
[9, 2, 4, 10, 1, 6, 8, 3, 7, 5]
[9, 2, 4, 10, 1, 6, 8, 5, 3, 7]
[9, 2, 4, 10, 1, 6, 8, 7, 3, 5]
[9, 2, 4, 10, 1, 8, 6, 3, 7, 5]
[9, 2, 4, 10, 1, 8, 6, 7, 3, 5]
[9, 2, 4, 10, 3, 5, 7, 8, 1, 6]
[9, 2, 4, 10, 5, 3, 7, 8, 1, 6]
[9, 2, 4, 10, 6, 1, 8, 7, 3, 5]
[9, 2, 6, 7, 3, 8, 4, 10, 1, 5]
[9, 2, 6, 7, 8, 3, 4, 10, 1, 5]
[9, 2, 7, 6, 3, 8, 4, 10, 1, 5]
[9, 2, 7, 6, 8, 3, 4, 10, 1, 5]
[9, 3, 4, 8, 6, 2, 7, 10, 1, 5]
[9, 3, 4, 8, 7, 2, 6, 10, 1, 5]
[9, 3, 5, 7, 8, 1, 6, 10, 2, 4]
[9, 4, 2, 10, 3, 5, 7, 8, 1, 6]
[9, 4, 2, 10, 5, 3, 7, 8, 1, 6]
[9, 4, 2, 10, 6, 1, 8, 7, 3, 5]
[9, 4, 2, 10, 6, 1, 8, 7, 5, 3]
[9, 4, 3, 8, 6, 2, 7, 10, 1, 5]
[9, 4, 3, 8, 7, 2, 6, 10, 1, 5]
[9, 5, 1, 10, 4, 3, 8, 6, 2, 7]
[9, 5, 1, 10, 4, 3, 8, 6, 7, 2]
[9, 5, 1, 10, 4, 3, 8, 7, 2, 6]
[9, 5, 1, 10, 4, 3, 8, 7, 6, 2]
[9, 5, 1, 10, 4, 8, 3, 6, 7, 2]
[9, 5, 1, 10, 4, 8, 3, 7, 6, 2]
[9, 5, 1, 10, 6, 2, 7, 8, 3, 4]
[9, 5, 1, 10, 6, 2, 7, 8, 4, 3]
[9, 5, 1, 10, 7, 2, 6, 8, 3, 4]
[9, 5, 1, 10, 7, 2, 6, 8, 4, 3]
[9, 5, 3, 7, 6, 8, 1, 10, 4, 2]
[9, 5, 3, 7, 8, 1, 6, 10, 2, 4]
[9, 5, 3, 7, 8, 1, 6, 10, 4, 2]
[9, 5, 3, 7, 8, 6, 1, 10, 4, 2]
[9, 5, 7, 3, 6, 8, 1, 10, 4, 2]
[9, 5, 7, 3, 8, 6, 1, 10, 4, 2]
[9, 6, 1, 8, 7, 3, 5, 10, 2, 4]
[9, 6, 1, 8, 7, 3, 5, 10, 4, 2]
[9, 6, 1, 8, 7, 5, 3, 10, 2, 4]
[9, 6, 1, 8, 7, 5, 3, 10, 4, 2]
[9, 6, 2, 7, 8, 3, 4, 10, 1, 5]
[9, 6, 2, 7, 8, 3, 4, 10, 5, 1]
[9, 6, 2, 7, 8, 4, 3, 10, 5, 1]
[9, 7, 2, 6, 8, 3, 4, 10, 1, 5]
[9, 7, 2, 6, 8, 3, 4, 10, 5, 1]
[9, 7, 2, 6, 8, 4, 3, 10, 5, 1]
[9, 7, 3, 5, 8, 6, 1, 10, 4, 2]
[9, 7, 5, 3, 8, 6, 1, 10, 4, 2]
[10, 1, 5, 9, 2, 6, 7, 3, 8, 4]
[10, 1, 5, 9, 2, 6, 7, 8, 3, 4]
[10, 1, 5, 9, 2, 7, 6, 3, 8, 4]
[10, 1, 5, 9, 2, 7, 6, 8, 3, 4]
[10, 1, 5, 9, 3, 4, 8, 6, 2, 7]
[10, 1, 5, 9, 3, 4, 8, 7, 2, 6]
[10, 1, 5, 9, 4, 3, 8, 6, 2, 7]
[10, 1, 5, 9, 4, 3, 8, 7, 2, 6]
[10, 1, 5, 9, 6, 2, 7, 8, 3, 4]
[10, 1, 5, 9, 7, 2, 6, 8, 3, 4]
[10, 1, 6, 8, 3, 5, 7, 9, 2, 4]
[10, 1, 6, 8, 3, 7, 5, 9, 2, 4]
[10, 1, 6, 8, 5, 3, 7, 9, 2, 4]
[10, 1, 6, 8, 7, 3, 5, 9, 2, 4]
[10, 1, 8, 6, 3, 7, 5, 9, 2, 4]
[10, 1, 8, 6, 7, 3, 5, 9, 2, 4]
[10, 2, 4, 9, 3, 5, 7, 8, 1, 6]
[10, 2, 4, 9, 5, 3, 7, 8, 1, 6]
[10, 2, 4, 9, 6, 1, 8, 7, 3, 5]
[10, 2, 4, 9, 6, 1, 8, 7, 5, 3]
[10, 3, 4, 8, 6, 2, 7, 9, 1, 5]
[10, 3, 4, 8, 7, 2, 6, 9, 1, 5]
[10, 3, 5, 7, 8, 1, 6, 9, 2, 4]
[10, 3, 5, 7, 8, 1, 6, 9, 4, 2]
[10, 4, 2, 9, 5, 3, 7, 6, 8, 1]
[10, 4, 2, 9, 5, 3, 7, 8, 1, 6]
[10, 4, 2, 9, 5, 3, 7, 8, 6, 1]
[10, 4, 2, 9, 5, 7, 3, 6, 8, 1]
[10, 4, 2, 9, 5, 7, 3, 8, 6, 1]
[10, 4, 2, 9, 6, 1, 8, 7, 3, 5]
[10, 4, 2, 9, 6, 1, 8, 7, 5, 3]
[10, 4, 2, 9, 7, 3, 5, 8, 6, 1]
[10, 4, 2, 9, 7, 5, 3, 8, 6, 1]
[10, 4, 3, 8, 6, 2, 7, 9, 1, 5]
[10, 4, 3, 8, 6, 2, 7, 9, 5, 1]
[10, 4, 3, 8, 6, 7, 2, 9, 5, 1]
[10, 4, 3, 8, 7, 2, 6, 9, 1, 5]
[10, 4, 3, 8, 7, 2, 6, 9, 5, 1]
[10, 4, 3, 8, 7, 6, 2, 9, 5, 1]
[10, 4, 8, 3, 6, 7, 2, 9, 5, 1]
[10, 4, 8, 3, 7, 6, 2, 9, 5, 1]
[10, 5, 1, 9, 6, 2, 7, 8, 3, 4]
[10, 5, 1, 9, 6, 2, 7, 8, 4, 3]
[10, 5, 1, 9, 7, 2, 6, 8, 3, 4]
[10, 5, 1, 9, 7, 2, 6, 8, 4, 3]
[10, 5, 3, 7, 8, 1, 6, 9, 2, 4]
[10, 5, 3, 7, 8, 1, 6, 9, 4, 2]
[10, 6, 1, 8, 7, 3, 5, 9, 2, 4]
[10, 6, 1, 8, 7, 3, 5, 9, 4, 2]
[10, 6, 1, 8, 7, 5, 3, 9, 4, 2]
[10, 6, 2, 7, 8, 3, 4, 9, 5, 1]
[10, 6, 2, 7, 8, 4, 3, 9, 5, 1]
[10, 7, 2, 6, 8, 3, 4, 9, 5, 1]
[10, 7, 2, 6, 8, 4, 3, 9, 5, 1]
   Админ: ой, мамочки!

евгений 2013-04-05 11:54:53 пишет:
[1, 5, 9, 2, 6, 7, 3, 8, 4, 10]
   Админ: верно

KaSeAl 2013-04-04 20:05:17 пишет:
Из возможного нашел такое: 3.8.6.1.10.4.2.9.5.7 Минимальное 15.
   Админ: верно

ivana2000 2013-04-04 19:53:11 пишет:
А если так:\n
1,10,4,3,8,6,2,7,9,5 ?
   Админ: верно

евгений 2013-04-04 19:21:01 пишет:
to ivana2000

а где Вы потеряли двойку ?

ivana2000 2013-04-04 19:07:19 пишет:
Легко показать, что подобная сумма не более 15
(Нужно сложить девять неравенств и учесть, что сумма всех чисел равна 55).\n\n
Далее, перебором находим:\n\n
1,9,10,3,7,5,4,6,8
   Админ: двойка где? :)

евгений 2013-04-04 15:28:21 пишет:
Хотя стоп,
ряд
2 10 1 9 3 8 4 7 5 6
дает нам 5+6+2 = 13
   Админ: Ну пусть будет 13. Кто больше? :)

евгений 2013-04-04 15:18:23 пишет:
почему ж не выходит?
числа то записаны по кругу, а потом
5+6+1 =12

евгений 2013-04-02 22:50:26 пишет:
12

для того, чтоб минимальная сумма была максимальной , все суммы должны быть как можно больше, это достигается путем чередования минимальных чисел ряда с максимальными:
1 10 2 9 3 8 4 7 5 6
   Админ: Надо складывать по три числа, из Вашего ряда Ваш ответ не выходит.

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