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1987.国际奥林匹克IMO.6数论——素数与合数证明问题。

2023-08-28 05:57 作者:封包  | 我要投稿

每一个不曾起舞的日子，都是对生命的辜负。

——《查拉图斯特拉如是说》尼采

%7B%5CHuge%20%0A%5Cbegin%7Barray%7D%7Bc%7D%0A(1987.IMO.2)%E5%B7%B2%E7%9F%A5%E6%AD%A3%E6%95%B4%E6%95%B0n%E2%89%A52%EF%BC%8C%E4%B8%94%E5%AF%B9%E4%BA%8E%E6%95%B4%E6%95%B0k%5C%5C%0A0%E2%89%A4k%E2%89%A4%5Csqrt%5B%5D%7B%5Cfrac%7Bn%7D%7B3%7D%20%7D%2Ck%5E2%2Bk%2Bn%E9%83%BD%E6%98%AF%E7%B4%A0%E6%95%B0%E3%80%82%5C%5C%0A%E8%AF%81%E6%98%8E%3A%E5%AF%B9%E4%BA%8E%E6%95%B4%E6%95%B0k%2C0%E2%89%A4k%E2%89%A4n-2%EF%BC%8Ck%5E2%2Bk%2Bn%E4%B9%9F%E6%98%AF%E7%B4%A0%E6%95%B0%E3%80%82%0A%5Cend%7Barray%7D%7D%20

%7B%5Clarge%20%0A%5Cbegin%7Barray%7D%7Bc%7D%0A%E8%80%83%E8%99%91%E5%8F%8D%E8%AF%81%E6%B3%95%EF%BC%9A%E8%8B%A5%E5%AD%98%E5%9C%A8m%2Cm%E2%89%A4n-2%5C%5C%0A%E4%BD%BF%E5%AD%98%E5%9C%A8%E7%B4%A0%E6%95%B0p%2Cp%7Cm%5E2%2Bm%2Bn%2C%E4%B8%94m%2Cp%E4%B8%BA%E5%85%B6%E4%B8%AD%E6%9C%80%E5%B0%8F%E7%9A%84%E6%95%B0%E3%80%82%5C%5C%0A%E2%88%B4m%5E2%2Bm%2Bn%E2%89%A5p%5E2%5C%5C%0Ap%7C(m-p)%5E2%2B(m-p)%2Bn%EF%BC%8C%E8%8B%A5m%E2%89%A5p%3A%5C%5C%0A%E5%88%99(m-p)%E6%98%BE%E7%84%B6%E4%B8%BA%E6%9B%B4%E5%B0%8F%E7%9A%84%E8%A7%A3%EF%BC%8C%E4%B8%8Em%E6%9C%80%E5%B0%8F%E6%80%A7%E7%9F%9B%E7%9B%BE%E3%80%82%5C%5C%0A%E2%88%B4m%E2%89%A4p-1%5C%5C%0Ap%7C(p-1-m)%5E2%2B(p-1-m)%2Bn%5C%5C%0A%E2%88%B5m%E6%98%AF%E6%9C%80%E5%B0%8F%E8%A7%A3%5C%5C%0A%E2%88%B4p-m-1%E2%89%A5m%5C%5C%0A%E2%88%B4p%E2%89%A52m%2B1%5C%5C%0A%E4%BB%A3%E5%85%A5%E5%9B%9Em%5E2%2Bm%2Bn%E2%89%A5p%5E2%EF%BC%9A%5C%5C%0A4m%5E2%2B4m%2B1%E2%89%A4m%5E2%2Bm%2Bn%3A%5C%5C%0An%2B3m%2B1%3C3m%5E2%2B3m%2B1%E2%89%A4n%E7%9F%9B%E7%9B%BE%EF%BC%81%5C%5C%0A%E6%95%85k%5E2%2Bk%2Bn%E4%B9%9F%E6%98%AF%E7%B4%A0%E6%95%B0%E3%80%82%0A%5Cend%7Barray%7D%7D%20

重点是能判断m^2+m+n≥p^2这一步放缩，使不等式更加紧凑容易导出矛盾。

高中生自行整理，如有错误，欢迎指出。

标签：数学

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