1998.国际奥林匹克IMO.4数论
封包
编辑于 2023年08月26日 01:43

认识你自己。

——古希腊哲学家 苏格拉底

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%7B%5Clarge%20%0A%5Cbegin%7Barray%7D%7Bc%7D%0A%E8%8B%A5ab%5E2%2Bb%2B7%7Ca%5E2b%2Ba%2Bb%5C%5C%0A%E8%AE%BEab%5E2%2Bb%2B7%3Dk%5C%5C%0Aa%5E2b%2Ba%2Bb%5Cequiv%200%20%5Cmod%7Bk%7D%5C%5C%0Aab%5E2%2Bb%2B7%5Cequiv%200%20%5Cmod%7Bk%7D%5C%5C%0Ab(a%5E2b%2Ba%2Bb)-a(ab%5E2%2Bb%2B7)%5Cequiv%200%20%5Cmod%7Bk%7D%5C%5C%0Ab%5E2-7a%20%5Cequiv%200%20%5Cmod%7Bk%7D%5C%5C%0A%E6%89%80%E4%BB%A5%7Cb%5E2-7a%7C%E2%89%A5ab%5E2%2Bb%2B7%5C%5C%0A(I)%E8%8B%A5b%5E2%3D7a%3A%5C%5C%0Aab%5E2%2Bb%2B7%7C0%E6%81%92%E6%88%90%E7%AB%8B%EF%BC%8C%E8%80%8C7%7Cb%5E2%EF%BC%8C%E8%AE%BEb%3D7k%5C%5C%0A%E5%88%99a%3D7k%5E2%5C%5C%0A%E9%82%A3%E4%B9%88(7k%5E2%2C7k)%2Ck%5Cin%20%5Cmathbb%7BZ%7D%E4%B8%BA%E4%B8%80%E7%BB%84%E8%A7%A3%E3%80%82%5C%5C%0A(II)%E8%8B%A5b%5E2%E2%89%A57a%2B1%3A%5C%5C%0Ab%5E2%EF%BC%9Eb%5E2-7a%E2%89%A5ab%5E2%2Bb%2B7%EF%BC%9Eab%5E2%E7%9F%9B%E7%9B%BE%E3%80%82%5C%5C%0A(III)%E8%8B%A5b%5E2%E2%89%A47a-1%3A%5C%5C%20%0A7a%3E7a-b%5E2%E2%89%A5ab%5E2%2Bb%2B7%3Eab%5E2%5C%5C%0Ab%5E2%3C7%5C%5C%0A%E2%88%B4b%3D1%E6%88%962%5C%5C%0Ab%3D1%E6%97%B6%EF%BC%9A%5C%5C%0Aa%2B8%7Ca%5E2%2Ba%2B1%5C%5C%0Aa%2B8%7Ca%2B8%5C%5C%0A%E2%88%B4a%2B8%7Ca%5E2%2Ba%2B1-a(a%2B8)%3D1-7a%5C%5C%0A%E2%88%B4a%2B8%7C7a-1%5C%5C%0A%E6%89%80%E4%BB%A5a%2B8%7C7a-1-7(a%2B8)%5C%5C%0Aa%2B8%7C57%3D3%C3%9719%5C%5C%0A%E2%88%B4a%3D11%E6%88%9649%5C%5C%0A%E5%BD%93b%3D2%E6%97%B6%EF%BC%9A%E4%BB%A3%E5%85%A5%E5%BE%97%E7%9F%9B%E7%9B%BE%5C%5C%0A%E7%BB%BC%E4%B8%8A%EF%BC%8C%E7%AC%A6%E5%90%88%E5%BE%97%E6%AD%A3%E6%95%B4%E6%95%B0%E5%AF%B9%E4%B8%BA(7k%5E2%2C7k)%2C(11%2C1)(49%2C1)%5C%5C%0A%5Cend%7Barray%7D%7D%20

imo中的送分题，把a^2b消去再进行适当放缩即可。

高中生自行整理，如有错误，欢迎指出。

数学
cv25998631