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1994.IMO.4數(shù)論——對稱性討論與整除更緊的放縮。

2023-08-26 18:16 作者:封包  | 我要投稿

我思故我在。

——唯理論哲學家 笛卡爾

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%7B%5Clarge%20%5Cbegin%7Barray%7D%7Bc%7D%0A%E6%AD%A4%E9%A2%98%E6%8A%80%E5%B7%A7%E6%80%A7%E6%AF%94%E8%BE%83%E9%AB%98%EF%BC%8C%5C%5C%E8%80%83%E8%99%91%E5%AF%B9%E7%A7%B0%E6%80%A7%E4%BB%A5%E5%8F%8A%E5%88%A9%E7%94%A8%E6%95%B4%E9%99%A4%E7%9A%84%E6%80%A7%E8%B4%A8%E6%9D%A5%E8%BF%9B%E8%A1%8C%E6%9B%B4%E7%B4%A7%E5%87%91%E7%9A%84%E6%94%BE%E7%BC%A9%E3%80%82%5C%5C%0A%E5%91%BD%E9%A2%98%E8%BD%AC%E5%8C%96%E4%B8%BAmn-1%7Cn%5E3%2B1%5C%5C%0Amn-1%7Cm(n%5E3%2B1)-n%5E2(mn-1)%3Dn%5E2%2Bm%5C%5C%0A%E2%88%B4n%5E2%2Bm%5Cequiv%200%20%5Cmod%7Bmn-1%7D%5C%5C%0Amn%5Cequiv%201%5Cmod%7Bmn-1%7D%20%5C%5C%0Am%5E2n%5Cequiv%20m%20%5Cmod%7Bmn-1%7D%5C%5C%0Am%5E2n%5Cequiv%20-n%5E2%20%5Cmod%7Bmn-1%7D%5C%5C%0A%E5%8F%88(n%2Cmn-1)%3D1%5C%5C%0A%E6%89%80%E4%BB%A5m%5E2%5Cequiv%20-n%20%5Cmod%7Bmn-1%7D%5C%5C%0A%E6%89%80%E4%BB%A5mn-1%7Cm%5E2%2Bn%EF%BC%8C%E6%81%B0%E5%A5%BD%E5%AF%B9%E7%A7%B0%5C%5C%5C%5C%0A%E4%B8%8D%E5%A6%A8%E5%85%88%E8%AE%A8%E8%AE%BAmn-1%7Cn%5E2%2Bm%5C%5C%0A(I).mn-1%3Dm%2Bn%5E2%5C%5C%0A%5Cfrac%7Bn%5E2%2B1%7D%7Bn-1%7D%3Dm%5Cin%20%5Cmathbb%7BZ%7D%5C%5C%0An-1%7Cn%5E2%2B1%E6%98%93%E5%BE%97n%3D2%E6%88%963%3A%5C%5C%0A%E4%BB%A3%E5%85%A5%E5%BE%97(1%2C2)(2%2C2)(5%2C2)(1%2C3)(5%2C3)5%E7%BB%84%E6%95%B0%E7%BB%84%5C%5C%5C%5C%0A(II).m%2Bn%5E2%E2%89%A52(mn-1)%5C%5C%0A%E8%A7%A3%E9%87%8A%E4%B8%80%E4%B8%8B%EF%BC%8C%E8%BF%99%E9%87%8CRHS%E4%B9%8B%E6%89%80%E4%BB%A5%E4%B9%98%E4%BB%A5%E4%BA%86%E4%B8%80%E4%B8%AA2%EF%BC%8C%5C%5C%E6%98%AF%E5%9B%A0%E4%B8%BA%E6%95%B4%E9%99%A4%E4%B8%AD%E8%8B%A5%E8%A2%AB%E9%99%A4%E6%95%B0%E5%92%8C%E9%99%A4%E6%95%B0%E4%B8%8D%E7%9B%B8%E7%AD%89%5C%5C%0A%E4%B8%94%E8%A2%AB%E9%99%A4%E6%95%B0%E4%B8%8D%E4%B8%BA0%EF%BC%8C%E9%82%A3%E4%B9%88%E6%98%BE%E7%84%B6%E8%A2%AB%E9%99%A4%E6%95%B0%E6%98%AF%E9%99%A4%E6%95%B0%E7%9A%84%E8%87%B3%E5%B0%912%E5%80%8D%E3%80%82%5C%5C%0A%E8%BF%99%E9%87%8C%E4%B8%A4%E8%BE%B9%E5%AF%B9%E4%BA%8Em%2Cn%E6%9D%A5%E8%AF%B4%E9%83%BD%E4%B8%8D%E6%98%AF%E9%BD%90%E6%AC%A1%E5%BC%8F%E5%AD%90%EF%BC%8C%E6%94%BE%E7%BC%A9%E8%BE%83%E9%BA%BB%E7%83%A6%E3%80%82%5C%5C%0A%E4%B8%8D%E5%A6%A8%E8%AE%BEm%E2%89%A5n(%E5%9B%A0%E4%B8%BA%E5%90%8E%E9%9D%A2%E6%88%91%E4%BB%AC%E5%8F%AF%E4%BB%A5%E5%AF%B9%E7%A7%B0%E5%9B%9E%E6%9D%A5%E5%89%A9%E4%BD%99%E8%A7%A3)%3A%5C%5C%0A%E5%BE%97%E5%88%B0n%5E2-n-2%E2%89%A40%2C%E5%BE%97%E5%88%B0n%3D1%E6%88%962%5C%5C%0A%E4%BB%A3%E5%85%A5%E5%BE%97%E5%88%B0(2%2C1)(3%2C1)%5C%5C%5C%5C%0A%E5%AF%B9%E4%BA%8Emn-1%7Cm%5E2%2Bn%E6%88%91%E4%BB%AC%E7%94%B1%E5%AF%B9%E7%A7%B0%E6%80%A7%EF%BC%8C%5C%5C%E5%B9%B6%E4%B8%94%E5%9C%A8%E7%AC%AC%E4%BA%8C%E6%AD%A5%E8%AE%BEn%E2%89%A5m%E5%8F%AF%E4%BB%A5%E5%BE%97%E5%88%B0%E5%AE%8C%E5%85%A8%E5%AF%B9%E7%A7%B0%E7%9A%84%E7%AD%94%E6%A1%88%5C%5C%0A%E7%BB%BC%E4%B8%8A(m%2Cn)%3D(1%2C2)(2%2C2)(5%2C2)(1%2C3)(5%2C3)(2%2C1)%5C%5C(3%2C1)%E5%92%8C%E5%AE%83%E4%BB%AC%E7%9A%84%E5%AF%B9%E7%A7%B0%3A(2%2C5)(3%2C5)%E5%85%B19%E7%BB%84%E7%AD%94%E6%A1%88%E3%80%82%5C%5C%0A%5Cend%7Barray%7D%7D%20

如果不使用對稱性在一開頭得到mn-1|m^2+n得到對稱性的話,在(II)處討論答案將十分吃力,巧用題目中的對稱性可以抵消非常多的計算量。個人覺得此處的對稱性處理技巧性十分的高,對數(shù)感要求很高。

高中生自行整理,如有錯誤,歡迎指出。

1994.IMO.4數(shù)論——對稱性討論與整除更緊的放縮。的評論 (共 條)

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