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Консультация № 188947
16.03.2016, 14:18
0.00 руб.
16.03.2016, 15:21
0
4
1
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Клетки квадратной таблицы 15х15 раскрашены в красный, синий и зеленый цвета. Докажите, что найдутся по крайней мере две строки, в которых клеток хотя бы одного цвета поровну.
Клетки квадратной таблицы 15х15 раскрашены в красный, синий и зеленый цвета. Докажите, что найдутся по крайней мере две строки, в которых клеток хотя бы одного цвета поровну.
Обсуждение
16.03.2016, 15:22
общий
Обратите, пожалуйста, внимание на эту консультацию, перенесённую из другого раздела.
Об авторе:
Facta loquuntur.
Facta loquuntur.
17.03.2016, 07:44
общий
Адресаты:
Если в двух строках отсутствует какой-нибудь цвет, например красный, считается, что в этих строках одинаковое количество красных клеток (ноль).
Количество клеток каждого цвета не должно быть меньше, чем
0+1+2+3+4+5+6+7+8+9+10+11+12+13+14=105.
Ясно, что клеток в таблице не хватит: 105+105+105 > 15*15.
http://problems.ru/view_problem_details_new.php?id=102844
Количество клеток каждого цвета не должно быть меньше, чем
0+1+2+3+4+5+6+7+8+9+10+11+12+13+14=105.
Ясно, что клеток в таблице не хватит: 105+105+105 > 15*15.
http://problems.ru/view_problem_details_new.php?id=102844
17.03.2016, 19:41
общий
это ответ
Здравствуйте, svrvsvrv!
В таблице всего 15*15=225 клеток. Если в каждой из двух строк содержится разное количество клеток одного цвета, то всего в таблице содержится, как минимум, 0+1+2+...+14=(1+14)*7=105 клеток одного цвета. То же верно для клеток двух других цветов. Значит, в таблице содержится, как минимум, 105*3=315 клеток. Но 225[$8800$]315. Полученное противоречие показывает, что в таблице найдутся по крайней мере две строки, в которых клеток хотя бы одного цвета поровну.
Как я понял, задача на принцип Дирихле. Излюбленная тема составителей олимпиадных задач.
С уважением.
В таблице всего 15*15=225 клеток. Если в каждой из двух строк содержится разное количество клеток одного цвета, то всего в таблице содержится, как минимум, 0+1+2+...+14=(1+14)*7=105 клеток одного цвета. То же верно для клеток двух других цветов. Значит, в таблице содержится, как минимум, 105*3=315 клеток. Но 225[$8800$]315. Полученное противоречие показывает, что в таблице найдутся по крайней мере две строки, в которых клеток хотя бы одного цвета поровну.
Как я понял, задача на принцип Дирихле. Излюбленная тема составителей олимпиадных задач.
С уважением.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
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