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2003.IMO.2数论——无穷递降法的整除性问题应用-放缩。

2023-08-26 18:51 作者:封包  | 我要投稿

人无法两只脚踏入同一条河流。

——古希腊哲学家 赫拉克利特

%7B%5CHuge%20%0A%5Cbegin%7Barray%7D%7Bc%7D%0A(2003.IMO.2)%E6%B1%82%E6%89%80%E6%9C%89%E7%9A%84%E6%AD%A3%E6%95%B4%E6%95%B0%E5%AF%B9(a%2Cb)%5C%5C%0A%E4%BD%BF%E5%BE%97%5Cfrac%7Ba%5E2%7D%7B2ab%5E2-b%5E3%2B1%7D%5Cin%20%5Cmathbb%7BN%5E%2B%7D%20%0A%5Cend%7Barray%7D%7D%20

%7B%5Clarge%20%5Cbegin%7Barray%7D%7Bc%7D%0A%E5%8E%9F%E5%91%BD%E9%A2%98%E5%8D%B3%E4%B8%BA2ab%5E2-b%5E3%2B1%7Ca%5E2%5C%5C%0A%E5%AF%B9%E4%BA%8E%E8%BF%99%E6%A0%B7%E5%A4%8D%E6%9D%82%E7%9A%84%E9%9D%9E%E9%BD%90%E6%AC%A1%E5%BC%8F%EF%BC%8C%E6%88%91%E4%BB%AC%E5%BE%88%E9%9A%BE%E5%AF%B9%E5%85%B6%E5%A4%84%E7%90%86%5C%5C%0A%E5%B9%B6%E4%B8%94%E8%8B%A5%E5%AF%B9%E5%85%B6%E8%BF%9B%E8%A1%8C%E4%BC%B0%E5%80%BC%EF%BC%8C%E4%B9%9F%E9%9A%BE%E4%BB%A5%E6%89%BE%E5%88%B0%E4%B8%80%E4%B8%AA%E7%9F%9B%E7%9B%BE%E7%82%B9%E3%80%82%5C%5C%0A%E8%BF%99%E9%87%8C%E4%BD%BF%E7%94%A8%E7%B1%BB%E4%BC%BC(1988.IMO.6)%E6%97%A0%E7%A9%B7%E9%80%92%E9%99%8D%E6%B3%95%E7%9A%84%E6%80%9D%E6%83%B3%E8%AE%A9%E6%88%91%E4%BB%AC%E6%89%BE%E5%88%B0%E8%90%BD%E8%84%9A%E7%82%B9%E3%80%82%5C%5C%5C%5C%0A%E8%AE%BE%5Cfrac%7Ba%5E2%7D%7B2ab%5E2-b%5E3%2B1%7D%3Dk%5C%5C%0Aa%5E2-2kb%5E2a%2Bb%5E3k-k%3D0%5C%5C%0A%E7%94%B1%E9%9F%A6%E8%BE%BE%E5%AE%9A%E7%90%86%3A%5C%5C%0Aa%2Ba'%3D%202kb%5E2%5C%5C%0Aaa'%3Db%5E3k-k%5C%5C%0A%E9%A6%96%E5%85%88a'%3D2kb%5E2-a%5Cin%20%5Cmathbb%7BZ%7D%0A%E4%B8%8D%E5%A6%A8%E8%AE%BEa%E2%89%A5a'%3A%5C%5C%0A%E9%82%A3%E4%B9%882kb%5E2%E2%89%A42a%E5%BE%97%E5%88%B0a%E2%89%A5b%5E2k%5C%5C%0Aa'%E2%89%A4%5Cfrac%7Bb%5E3k-k%7D%7Bb%5E2k%7D%3Db-%5Cfrac%7B1%7D%7Bb%5E2%7D%3Cb%5C%5C%0A%E6%88%91%E4%BB%AC%E7%9F%A5%E9%81%93(a'%2Cb)%E6%98%AF%E4%B8%8D%E5%AE%9A%E6%96%B9%E7%A8%8B%E7%9A%84%E5%8F%A6%E4%B8%80%E7%BB%84%E8%A7%A3%EF%BC%9A%5C%5C%0A2a'b%5E2-b%5E3%2B1%7Ca'%5E2%5C%5C%0A%E6%89%80%E4%BB%A5b%5E2%3Ea'%5E2%E2%89%A52a'b%5E2-b%5E3%2B1%3Db%5E2(2a'-b)%2B1%5C%5C%0A%E6%89%80%E4%BB%A52a'-b%3C1%5C%5C%0A%E8%80%8Ck%E6%98%AF%E4%B8%80%E4%B8%AA%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%8C%E6%89%80%E4%BB%A52a'-b%E2%89%A50%5C%5C%0A(%E6%B3%A8%EF%BC%9A%E7%94%A8%E9%9F%A6%E8%BE%BE%E5%AE%9A%E7%90%86%E4%B8%ADa%2Ca'%E7%9A%84%E5%85%B3%E7%B3%BB%E5%90%8C%E6%A0%B7%E5%8F%AF%E5%BE%972a'-b%E2%89%A50)%5C%5C%0A%E9%82%A3%E4%B9%882a'%3Db%2C%E8%AE%BEb%3D2l%E5%BE%97%3Aa%3D8l%5E4-l%2Ca'%3Dl(b%3D2l)%5C%5C%0A%E8%8B%A5a'%3D0%3Ab%3D1%2Ca%3D2l%5C%5C%5C%5C%0A%E7%BB%BC%E4%B8%8A(a%2Cb)%3D(l%2C2l)(2l%2C1)(8l%5E4-l%2Cl)%2Cl%5Cin%20%5Cmathbb%7BN%5E%2B%7D%0A%5Cend%7Barray%7D%7D%20

本题直接硬解对计算能力要求极高，用韦达跳跃得到一定关系后再解不等式能够更加清晰简洁。

高中生自行整理，如有错误，欢迎指出。

标签：数学

2003.IMO.2数论——无穷递降法的整除性问题应用-放缩。的评论 (共 条)

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