{"id":328,"date":"2012-03-01T01:32:31","date_gmt":"2012-02-29T16:32:31","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=328"},"modified":"2021-11-04T14:50:31","modified_gmt":"2021-11-04T05:50:31","slug":"htb200805","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb200805\/","title":{"rendered":"\u4e00\u6a4b\u59272008\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(n\\) \u3092 \\(3\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b.\r\n\\(2n\\) \u679a\u306e\u30ab\u30fc\u30c9\u304c\u3042\u308a, \u305d\u306e\u3046\u3061\u8d64\u3044\u30ab\u30fc\u30c9\u306e\u679a\u6570\u306f \\(6\\) , \u767d\u3044\u30ab\u30fc\u30c9\u306e\u679a\u6570\u306f \\(2n-6\\) \u3067\u3042\u308b. \u3053\u308c\u3089 \\(2n\\) \u679a\u306e\u30ab\u30fc\u30c9\u3092, \u7bb1 A \u3068\u7bb1 B \u306b \\(n\\) \u679a\u305a\u3064\u7121\u4f5c\u70ba\u306b\u5165\u308c\u308b. \\(2\\) \u3064\u306e\u7bb1\u306e\u5c11\u306a\u304f\u3068\u3082\u4e00\u65b9\u306b\u8d64\u3044\u30ab\u30fc\u30c9\u304c\u3061\u3087\u3046\u3069 \\(k\\) \u679a\u5165\u3063\u3066\u3044\u308b\u78ba\u7387\u3092 \\(p _ k\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(p _ 2\\) \u3092 \\(n\\) \u306e\u5f0f\u3067\u8868\u305b. \u3055\u3089\u306b, \\(p _ 2\\) \u3092\u6700\u5927\u306b\u3059\u308b \\(n\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(p _ 1 +p _ 2 \\lt p _ 0 +p _ 3\\) \u3092\u307f\u305f\u3059 \\(n\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\(2n\\) \u679a\u306e\u30ab\u30fc\u30c9\u3092\u7bb1 A , \u7bb1 B \u306b \\(n\\) \u679a\u305a\u3064\u3044\u308c\u308b\u65b9\u6cd5\u306f,\r\n\u5168\u90e8\u3067 \\({} _ {2n} \\text{C} {} _ n\\) \u901a\u308a\u3042\u308b.
    \r\n\u3053\u306e\u3046\u3061, \u7bb1 A \u304b\u7bb1 B \u306b\u8d64\u3044\u30ab\u30fc\u30c9\u304c\u3061\u3087\u3046\u3069 \\(2\\) \u679a\u5165\u308b\u65b9\u6cd5\u306f\r\n\\[\r\n2 \\cdot {} _ {6} \\text{C} {} _ {2} \\cdot {} _ {2n-6} \\text{C} {} _ {n-2} \\quad \\text{\u901a\u308a}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\\begin{align}\r\np _ 2 & = \\dfrac{2 \\cdot {} _ {6} \\text{C} {} _ {2} \\cdot {} _ {2n-6} \\text{C} {} _ {n-2}}{{} _ {2n} \\text{C} {} _ n} \\\\\r\n& = \\dfrac{2 \\cdot 15 \\cdot (2n-6)! \\, n! \\, n!}{(n-2)! \\, (n-4)! \\, (2n)!} \\\\\r\n& = \\dfrac{30 n^2 (n-1)^2 (n-2) (n-3)}{2n (2n-1)(2n-2)(2n-3)(2n-4)(2n-5)} \\\\\r\n& =\\underline{\\dfrac{15 n(n-1)(n-3)}{4(2n-1)(2n-3)(2n-5)}}\r\n\\end{align}\\]\r\n\\(f(n) =p _ 2\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf(n+1) = \\dfrac{15 (n+1) n (n-2)}{4(2n+1)(2n-1)(2n-3)}\r\n\\]\r\n\u3053\u3053\u3067, \\(\\dfrac{f(n+1)}{f(n)}\\) \u3068 \\(1\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{f(n+1)}{f(n)} & = \\dfrac{(n+1)(n-2)(2n-5)}{(n-1)(n-3)(2n+1)} \\gt 1 \\\\\r\n2n^3-7n^2+n+10 & \\gt 2n^3-7n^2+2n+3 \\\\\r\n\\text{\u2234} \\quad n & \\lt 7\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(n) \\ ( n \\geqq 3 )\\) \u306e\u5927\u5c0f\u306f\r\n\\[\r\nf(3) \\lt f(4) \\lt \\cdots \\lt f(7) = f(8) \\gt f(9) \\gt \\cdots\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(n\\) \u306e\u5024\u306f\r\n\\[\r\nn =\\underline{7, 8}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\\begin{align}\r\np _ 0 & = \\dfrac{2 \\cdot {} _ {2n-6} \\text{C} {} _ {n}}{{} _ {2n} \\text{C} {} _ n} =\\dfrac{2 \\cdot (2n-6)! \\, n! \\, n!}{n! \\, (n-6)! \\, (2n)!} \\\\\r\n& = \\dfrac{2n(n-1)(n-2)(n-3)(n-4)(n-5)}{2n(2n-1)(2n-2)(2n-3)(2n-4)(2n-5)} \\\\\r\n& = \\dfrac{(n-3)(n-4)(n-5)}{4(2n-1)(2n-3)(2n-5)} , \\\\\r\np _ 1 & = \\dfrac{2 \\cdot {} _ {6} \\text{C} {} _ {1} \\cdot {} _ {2n-6} \\text{C} {} _ {n-1}}{{} _ {2n} \\text{C} {} _ n} =\\dfrac{12 \\cdot (2n-6)! \\, n! \\, n!}{(n-1)! \\, (n-5)! \\, (2n)!} \\\\\r\n& = \\dfrac{12n^2(n-1)(n-2)(n-3)(n-4)}{2n(2n-1)(2n-2)(2n-3)(2n-4)(2n-5)} \\\\\r\n& = \\dfrac{3n(n-3)(n-4)}{2(2n-1)(2n-3)(2n-5)} , \\\\\r\np _ 3 & = \\dfrac{{} _ {6} \\text{C} {} _ {3} \\cdot {} _ {2n-6} \\text{C} {} _ {n-3}}{{} _ {2n} \\text{C} {} _ n} = \\dfrac{20 \\cdot (2n-6)! \\, n! \\, n!}{(n-3)! \\, (n-3)! \\, (2n)!} \\\\\r\n& = \\dfrac{20n^2(n-1)^2(n-2)^2}{2n(2n-1)(2n-2)(2n-3)(2n-4)(2n-5)} \\\\\r\n& = \\dfrac{5n(n-1)(n-2)}{2(2n-1)(2n-3)(2n-5)}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\np _ 1 +p _ 2 & \\lt p _ 0 +p _ 3 \\\\\r\n6n(n-3) (n-4) +15n(n-1) & (n-3) \\\\\r\n\\lt (n-3)(n-4) & (n-5) +10n(n-1)(n-2) \\\\\r\n6n^3 -42n^2 +72n +15n^3 -60n^2 & +45n \\\\\r\n\\lt n^3 -12n^2 +47n & -60 +10n^3 -30n^2 +20n \\\\\r\n10n^3 -60n^2 +50n +60 & \\lt 0 \\\\\r\nn^3 -6n^2 +5n +6 & \\lt 0 \\\\\r\n(n-2)(n^2 -4n -3) & \\lt 0 \\\\\r\n(n-2) \\left\\{ n -\\left( 2-\\sqrt{7} \\right) \\right\\} \\left\\{ n -\\left( 2+\\sqrt{7} \\right) \\right\\} & \\lt 0 \\\\\r\n\\text{\u2234} \\quad 2 \\lt n \\lt 2+\\sqrt{7} &\r\n\\end{align}\\]\r\n\\(2 \\lt \\sqrt{7} \\lt 3\\) \u306a\u306e\u3067, \u6c42\u3081\u308b \\(n\\) \u306e\u5024\u306f\r\n\\[\r\nn = \\underline{3, 4}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092 \\(3\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b. \\(2n\\) \u679a\u306e\u30ab\u30fc\u30c9\u304c\u3042\u308a, \u305d\u306e\u3046\u3061\u8d64\u3044\u30ab\u30fc\u30c9\u306e\u679a\u6570\u306f \\(6\\) , \u767d\u3044\u30ab\u30fc\u30c9\u306e\u679a\u6570\u306f \\(2n-6\\) \u3067\u3042\u308b. \u3053\u308c\u3089 \\(2n\\) \u679a\u306e\u30ab\u30fc\u30c9\u3092, \u7bb1 A … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[57],"tags":[146,16],"class_list":["post-328","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2008","tag-hitotsubashi","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/328","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=328"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/328\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=328"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=328"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=328"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}