{"id":1290,"date":"2015-09-23T00:32:22","date_gmt":"2015-09-22T15:32:22","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1290"},"modified":"2021-09-16T09:49:11","modified_gmt":"2021-09-16T00:49:11","slug":"thr201506","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201506\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2015\uff1a\u7b2c6\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(k \\geqq 2\\) \u3068 \\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \\(n\\) \u304c \\(k\\) \u500b\u306e\u9023\u7d9a\u3059\u308b\u81ea\u7136\u6570\u306e\u548c\u3067\u3042\u308b\u3068\u304d, \u3059\u306a\u308f\u3061\r\n\\[\r\nn = m +(m+1) +\\cdots +(m+k-1)\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306a\u81ea\u7136\u6570 \\(m\\) \u304c\u5b58\u5728\u3059\u308b\u3068\u304d, \\(n\\) \u3092 \\(k -\\text{\u9023\u7d9a\u548c}\\) \u3068\u547c\u3076\u3053\u3068\u306b\u3059\u308b. \u305f\u3060\u3057, \u81ea\u7136\u6570\u3068\u306f \\(1\\) \u4ee5\u4e0a\u306e\u6574\u6570\u306e\u3053\u3068\u3067\u3042\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(n\\) \u304c \\(k -\\text{\u9023\u7d9a\u548c}\\) \u3067\u3042\u308b\u3053\u3068\u306f, \u6b21\u306e\u6761\u4ef6 (A) , (B) \u306e\u4e21\u65b9\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3068\u540c\u5024\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p>\r\n<ol>\r\n<li><p>(A)\u3000\\(\\dfrac{n}{k} -\\dfrac{k}{2} +\\dfrac{1}{2}\\) \u306f\u6574\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n<li><p>(B)\u3000\\(2n \\gt k^2\\) \u304c\u6210\u308a\u7acb\u3064.<\/p><\/li>\r\n<\/ol><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(f\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \\(n = 2^f\\) \u306e\u3068\u304d, \\(n\\) \u304c \\(k -\\text{\u9023\u7d9a\u548c}\\) \u3068\u306a\u308b\u3088\u3046\u306a\u81ea\u7136\u6570 \\(k \\geqq 2\\) \u306f\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(f\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \\(p\\) \u3092 \\(2\\) \u3067\u306a\u3044\u7d20\u6570\u3068\u3059\u308b. \\(n = p^f\\) \u306e\u3068\u304d, \\(n\\) \u304c \\(k -\\text{\u9023\u7d9a\u548c}\\) \u3068\u306a\u308b\u3088\u3046\u306a\u81ea\u7136\u6570 \\(k \\geqq 2\\) \u306e\u500b\u6570\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>(P)\uff1a\u300c \\(n\\) \u304c \\(k - \\text{\u9023\u7d9a\u548c}\\) \u3067\u3042\u308b. \u300d\u3068\u304a\u304f.<\/p>\r\n<ul>\r\n<li><p>\\((\\text{P}) \\Rightarrow (\\text{A}) , (\\text{B})\\) \u306e\u8a3c\u660e<br \/>\r\n(P) \u3088\u308a\r\n\\[\\begin{align}\r\nn & = mk +1 +2 + \\cdots +(n-1) \\\\\r\n& = mk +\\dfrac{(k-1) k}{2} \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n[1] \u306e\u4e21\u8fba\u3092 \\(k\\) \u3067\u5272\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{n}{k} & = m +\\dfrac{k-1}{2} \\\\\r\n\\text{\u2234} \\quad \\dfrac{n}{k} & -\\dfrac{k}{2} +\\dfrac{1}{2} = m \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, (A) \u304c\u6210\u7acb\u3059\u308b.<br \/>\r\n\u307e\u305f, [1] \u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n2n & = 2mk +k^2 -k \\\\\r\n& = k^2 +k (2m-1) \\\\\r\n& \\gt k^2 \\quad ( \\ \\text{\u2235} \\ 2m-1 \\gt 0 \\ ) \\ .\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, (B) \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<li><p>\\((\\text{A}) , (\\text{B}) \\Rightarrow (\\text{P})\\) \u306e\u8a3c\u660e<br \/>\r\n(A) \u3088\u308a, \\(\\dfrac{n}{k} -\\dfrac{k}{2} +\\dfrac{1}{2} = M\\) \uff08 \\(M\\) \u306f\u6574\u6570\uff09\u3068\u304a\u3051\u308b.<br \/>\r\n(B) \u3088\u308a, \\(\\dfrac{n}{k} \\gt \\dfrac{k}{2}\\) \u306a\u306e\u3067\r\n\\[\r\nM \\gt \\dfrac{1}{2} \\ .\r\n\\]\r\n\u3064\u307e\u308a, \\(M\\) \u306f\u81ea\u7136\u6570\u3067\u3042\u308b.<br \/>\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nn & -\\dfrac{(k-1) k}{2} = kM \\\\\r\n\\text{\u2234} \\quad n & = kM +1 +2 + \\cdots +(n-1) \\\\\r\n& = M +(M+1) +(M+2) + \\cdots +(M+n-1) \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, (P) \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ul>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\u80cc\u7406\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<br \/>\r\n\u6761\u4ef6\u3092\u6e80\u305f\u3059 \\(k \\geqq 2\\) \u304c\u3042\u308b\u306a\u3089\u3070, [1] \u3088\u308a\r\n\\[\\begin{align}\r\n2^f & = mk +\\dfrac{(k-1) k}{2} \\\\\r\n\\text{\u2234} \\quad 2^{f+1} & = k ( k +2m -1 ) \\ .\r\n\\end{align}\\]\r\n\\(2m -1\\) \u306f\u5947\u6570\u306a\u306e\u3067, \\(k\\) \u3068 \\(k +2m -1\\) \u306f\u5947\u5076\u304c\u7570\u306a\u308b.<br \/>\r\n\u307e\u305f, \\(k \\lt k +2m -1\\) \u306a\u306e\u3067\r\n\\[\r\nk = 1 , \\ k +2m -1 = 2^{f+1} \\ .\r\n\\]\r\n\u3053\u308c\u306f, \\(k \\geqq 2\\) \u3092\u6e80\u305f\u3055\u306a\u3044\u306e\u3067, \u4e0d\u9069.<br \/>\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n<p><strong>(3)<\/strong><\/p>\r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\np^f & = mk +\\dfrac{(k-1) k}{2} \\\\\r\n\\text{\u2234} \\quad 2 p^f & = k ( k +2m -1 ) \\ .\r\n\\end{align}\\]\r\n\\(k \\ ( \\geqq 2 )\\) \u3068 \\(k +2m -1 \\ ( \\geqq 3 )\\) \u306f\u5947\u5076\u304c\u7570\u306a\u308a, \\(p\\) \u304c\u5947\u6570\u306e\u7d20\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089\r\n\\[\\begin{align}\r\n( k , k+2m-1 ) & = \\left\\{ \\begin{array}{ll} \\left( p^g , 2 p^{f-g} \\right) & ( \\ k \\ \\text{\u304c\u5947\u6570\u306e\u3068\u304d} \\ ) \\\\\r\n\\left( 2 p^{g-1} , p^{f-g+1} \\right) & ( \\ k \\ \\text{\u304c\u5076\u6570\u306e\u3068\u304d} \\ ) \\end{array} \\right. \\\\\r\n& ( \\ \\text{\u305f\u3060\u3057} \\ g = 1 , 2 ,\\cdots , f ) \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u307f\u305f\u3059 \\(k\\) \u306f \\(2f\\) \u500b\u3042\u308b\u304c, \u3069\u306e\u5834\u5408\u3082 \\(k \\neq k +2m -1\\) \uff08 \u2235 \u5947\u5076\u304c\u7570\u306a\u308b. \uff09\u306a\u306e\u3067, \u3053\u306e\u3046\u3061 \\(k \\lt k +2m -1\\) \u3092\u307f\u305f\u3059\u3082\u306e\u306f, \\(\\dfrac{2f}{2} = f\\) \u500b\u3067\u3042\u308b.<br \/>\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u500b\u6570\u306f\r\n\\[\r\n\\underline{f \\quad \\text{\u500b}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(k \\geqq 2\\) \u3068 \\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \\(n\\) \u304c \\(k\\) \u500b\u306e\u9023\u7d9a\u3059\u308b\u81ea\u7136\u6570\u306e\u548c\u3067\u3042\u308b\u3068\u304d, \u3059\u306a\u308f\u3061 \\[ n = m +(m+1) +\\cdots +(m+k-1) \\] \u304c\u6210\u308a\u7acb\u3064\u3088 &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/thr201506\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[131],"tags":[148,137],"class_list":["post-1290","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2015","tag-tohoku_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1290","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=1290"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1290\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}