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趣味數學問題懸賞4 初三，高一組 參考答案

2022-12-06 13:16 作者:数学只因  | 我要投稿

不知道啥是趣味數學問題懸賞看這裏

第一題解答： 

%5Cbegin%7Baligned%7D%0A%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%2Ba%7D%2B%5Cfrac%7B1%7D%7Bx%2Ba%5E2%7D%26%3D0%5C%5C%0A(%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7B1%7D%7Bx%2Ba%7D%2B%5Cfrac%7B1%7D%7Bx%2Ba%5E2%7D)(x)(x%2Ba)(x%2Ba%5E2)%26%3D%200%20%5C%5C%0A(x%2Ba)(x%2Ba%5E2)%2Bx(x%2Ba%5E2)%2Bx(x%2Ba)%26%3D0%5C%5C%0Ax%5E2%2Ba%5E2x%2Bax%2Ba%5E3%2Bx%5E2%2Ba%5E2x%2Bx%5E2%2Bax%26%3D0%5C%5C%0A3x%5E2%2B2a%5E2x%2B2ax%2Ba%5E3%26%3D0%0A%5Cend%7Baligned%7D

用公式法可得：

x_1%3D%5Cfrac%7B-a%5E2-a%2B%5Csqrt%7Ba%5E4-a%5E3%2Ba%5E2%7D%7D%7B3%7D%5C%5C%0Ax_2%3D%5Cfrac%7B-a%5E2-a-%5Csqrt%7Ba%5E4-a%5E3%2Ba%5E2%7D%7D%7B3%7D

再做一些缩放：

%5Cbegin%7Baligned%7D%0A%26%5Cbecause%20a%3C0%5C%5C%0A%26%5Ctherefore%5C%5C%0A%26%5Cfrac%7B-a%5E2-a%2B%5Csqrt%7Ba%5E4-2a%5E3%2Ba%5E2%7D%7D%7B3%7D%3E%5Cfrac%7B-a%5E2-a%2B%5Csqrt%7Ba%5E4-a%5E3%2Ba%5E2%7D%7D%7B3%7D%3E%5Cfrac%7B-a%5E2-a%2B%5Csqrt%7Ba%5E4%2B2a%5E3%2Ba%5E2%7D%7D%7B3%7D%5C%5C%0A%26%5Cfrac%7B-a%5E2-a-%5Csqrt%7Ba%5E4%2B2a%5E3%2Ba%5E2%7D%7D%7B3%7D%3E%5Cfrac%7B-a%5E2-a-%5Csqrt%7Ba%5E4-a%5E3%2Ba%5E2%7D%7D%7B3%7D%3E%5Cfrac%7B-a%5E2-a-%5Csqrt%7Ba%5E4-2a%5E3%2Ba%5E2%7D%7D%7B3%7D%5C%5C%0A%26%5Ctherefore%5C%5C%0A%26%5Cfrac%7B-a%5E2-a%2B%7Ca%5E2-a%7C%7D%7B3%7D%3Ex_1%3E%5Cfrac%7B-a%5E2-a%2B%7Ca%5E2%2Ba%7C%7D%7B3%7D%5C%5C%0A%26%5Cfrac%7B-a%5E2-a-%7Ca%5E2%2Ba%7C%7D%7B3%7D%3Ex_2%3E%5Cfrac%7B-a%5E2-a-%7Ca%5E2-a%7C%7D%7B3%7D%5C%5C%0A%26%5Ctherefore%5C%5C%0A%26%5Cfrac%7B-a%5E2-a%2Ba%5E2-a%7D%7B3%7D%3Ex_1%3E%5Cfrac%7B-(a%5E2%2Ba)%2B%7Ca%5E2%2Ba%7C%7D%7B3%7D%5C%5C%0A%26%5Cfrac%7B-(a%5E2%2Ba)-%7Ca%5E2%2Ba%7C%7D%7B3%7D%3Ex_2%3E%5Cfrac%7B-a%5E2-a-a%5E2%2Ba%7D%7B3%7D%5C%5C%0A%26%5Ctherefore-%5Cfrac%7B2a%7D%7B3%7D%3Ex_1%3E0%2C0%3Ex_2%3E-%5Cfrac%7B2a%5E2%7D%7B3%7D%0A%0A%5Cend%7Baligned%7D


第二題参考思路：

对，你没看错，这里只给思路（狗头保命）

因为如果使用正规解答的话，学校会想为啥那么多人会超纲的定理

不说了，以下是思路：

首先，两边拆括号

再移项，把不等号的右式移成零

这时可以注意到左式是一个完全对称式

着代表着你可以把 x%2Cy%2Cz 进行排序

事情突然 简 ( fu ) 单 ( za ) 了起来

只要把左式分成几部分，且每部分都大于零就行了

主要是我太懒了，不想写过程，自己摸索吧

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