{"id":1975,"date":"2021-11-16T08:18:36","date_gmt":"2021-11-15T23:18:36","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1975"},"modified":"2021-11-16T08:18:36","modified_gmt":"2021-11-15T23:18:36","slug":"kyr202106","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr202106\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2021\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\u6b21\u306e\u5404\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. \u554f 1<\/strong>\u3000\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b.\r\n\\(3^n -2^n\\) \u304c\u7d20\u6570\u306a\u3089\u3070 \\(n\\) \u3082\u7d20\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. \u554f 2<\/strong>\u3000\\(a\\) \u3092 \\(1\\) \u3088\u308a\u5927\u304d\u3044\u5b9a\u6570\u3068\u3059\u308b.\r\n\u5fae\u5206\u53ef\u80fd\u306a\u95a2\u6570 \\(f(x)\\) \u304c \\(f(a) = a f(1)\\) \u3092\u6e80\u305f\u3059\u3068\u304d, \u66f2\u7dda \\(y = f(x)\\) \u306e\u63a5\u7dda\u3067\u539f\u70b9 \\(( 0 , 0 )\\) \u3092\u901a\u308b\u3082\u306e\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

    \u554f 1<\/strong><\/p>\r\n

    \u5bfe\u5076\u3092\u3068\u3063\u3066, \u300c \\(n\\) \u304c\u5408\u6210\u6570\u306a\u3089\u3070, \\(3^n -2^n\\) \u3082\u5408\u6210\u6570\u3067\u3042\u308b. \u300d... [A] \u3092\u793a\u305b\u3070\u3088\u3044.
    \r\n\\(n = ab\\) \uff08 \\(a , b\\) \u306f \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\uff09\u3068\u8868\u305b\u3066\r\n\\[\\begin{align}\r\n3^n -2^n & = \\left( 3^a \\right)^b -\\left( 2^a \\right)^b \\\\\r\n& = \\underline{\\left( 3^a -2^a \\right)} _ {[1]} \\underline{\\left( 3^{a(b-1)} +3^{a(b-2)} \\cdot 2^a +\\cdots +2^{a(b-1)} \\right)} _ {[2]}\r\n\\end{align}\\]\r\n[1] [2] \u3068\u3082\u306b \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u306a\u306e\u3067, \\(3^n -2^n\\) \u3082\u5408\u6210\u6570\u3067\u3042\u308b.
    \r\n\u3088\u3063\u3066, [A] \u304c\u793a\u3055\u308c, \u984c\u610f\u3082\u793a\u3055\u308c\u305f.<\/p>\r\n

    \u554f 2<\/strong><\/p>\r\n

    \\(x\\) \u5ea7\u6a19\u304c \\(t\\) \u3067\u3042\u308b\u70b9\u3067\u306e\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\r\ny = f'(t) (x-t) +f(t)\r\n\\]\r\n\u3053\u308c\u304c\u539f\u70b9\u3092\u901a\u308b\u306a\u3089\u3070\r\n\\[\\begin{align}\r\n0 & = -t f'(t) +f(t) \\\\\r\n\\text{\u2234} \\quad f(t) & = t f'(t) \\quad ... [1]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u3092\u307f\u305f\u3059 \\(t\\) \u306e\u5b58\u5728\u3092\u793a\u305b\u3070\u3088\u3044.
    \r\n\\(g(x) = \\dfrac{g(x)}{x} \\ (x \\neq 0)\\) \u3068\u304a\u304f\u3068, \u3053\u308c\u306f\u5b9a\u7fa9\u57df\u3067\u5fae\u5206\u53ef\u80fd\u306a\u95a2\u6570\u3067\u3042\u308a\r\n\\[\r\ng'(x) = \\dfrac{x f'(x) -f(x)}{x^2} \\quad ... [2]\r\n\\]\r\n\u5e73\u5747\u5024\u306e\u5b9a\u7406\u3088\u308a, \\(1 \\leqq c \\leqq a\\) \u304b\u3064\r\n\\[\\begin{align}\r\ng'(c) & = \\dfrac{g(a) -g(1)}{a-1} \\\\\r\n& = \\dfrac{f(a) -a f(1)}{a (a-1)} \\\\\r\n& = 0 \\quad ( \\text{\u2235} \\ f(a) = a f(1) )\r\n\\end{align}\\]\r\n\u3092\u307f\u305f\u3059 \\(c\\) \u304c\u5b58\u5728\u3059\u308b.
    \r\n[2] \u3092\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{c f'(c) -f(c)}{c^2} & = 0 \\\\\r\n\\text{\u2234} \\quad f(c) & = c f'(c)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(t=c\\) \u304c [1] \u3092\u307f\u305f\u3059\u306e\u3067, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u5404\u554f\u306b\u7b54\u3048\u3088. \u554f 1\u3000\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b. \\(3^n -2^n\\) \u304c\u7d20\u6570\u306a\u3089\u3070 \\(n\\) \u3082\u7d20\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b. \u554f 2\u3000\\(a\\) \u3092 \\(1\\) \u3088\u308a\u5927\u304d\u3044\u5b9a\u6570\u3068\u3059\u308b. \u5fae\u5206\u53ef … \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":[171],"tags":[140,165],"class_list":["post-1975","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2021","tag-kyoto_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1975","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=1975"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1975\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1975"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1975"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1975"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}