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[函数] 第二大的数

本帖最后由 hbghlyj 于 2019-7-11 00:46 编辑

用绝对值函数和初等函数的有限次复合表示四个实数a,b,c,d中第二大的数(有多种方法)
提示
a,b中第二大的数=$\frac{a+b-|a-b|}2$=$\frac{2ab}{a+b+|a-b|}$
a,b,c中第二大的数=$\frac{2a+2b+|a+b-|a-b|-2c|-|a+b+|a-b|-2c|}4$=$ - \frac{{256abc}}{{(\left| {a + b - 2c - \left| {a - b} \right|} \right| + \left| {a + b - 2c - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a + b - 2c - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|} \right| + \left| {a - b} \right| - a - b - 14c)(\left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|} \right| + \left| {a - b} \right| + a + b + 14c)}}$=$-\frac{8abc}{{(a-b)^2-4c(a+b)-\left| {a - b} \right|( - \left| {a + b - 2c - \left| {a - b} \right|} \right| + a + b - 2c)}}$

代码弄不好,干脆就贴图吧……

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回复 2# kuing
经过折腾终于弄好了

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我先提一种方法,设a,b,c,d从大到小排列为x,y,z,w,则x和w能用绝对值函数表示,a+b+c+d-x-w=y+z,abcd/xw=yz,然后求出这个二次方程的较大根即可.
最终结果繁琐惊人.就不贴了.

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$$ - \left( {\left( { - 6{a^3} - 18{a^2}b - 18a{b^2} - 6{b^3} - 28{a^2}c - 56abc - 28{b^2}c - 40a{c^2} - 40b{c^2} - 16{c^3} - 48{a^2}d - 96abd - 48{b^2}d - 128acd - 128bcd - 64{c^2}d - 96a{d^2} - 96b{d^2} - 64c{d^2} + 6a{{\left| {a - b} \right|}^2} + 6b{{\left| {a - b} \right|}^2} + 4c{{\left| {a - b} \right|}^2} + 5{a^2}\left| {a + b - 2c - \left| {a - b} \right|} \right| + 10ab\left| {a + b - 2c - \left| {a - b} \right|} \right| + 5{b^2}\left| {a + b - 2c - \left| {a - b} \right|} \right| + 12ac\left| {a + b - 2c - \left| {a - b} \right|} \right| + 12bc\left| {a + b - 2c - \left| {a - b} \right|} \right| + 4{c^2}\left| {a + b - 2c - \left| {a - b} \right|} \right| + 16ad\left| {a + b - 2c - \left| {a - b} \right|} \right| + 16bd\left| {a + b - 2c - \left| {a - b} \right|} \right| - 16{d^2}\left| {a + b - 2c - \left| {a - b} \right|} \right| + 6a\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right| + 6b\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right| + } \right.} \right.4c\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right| + {\left| {a - b} \right|^2}\left| {a + b - 2c - \left| {a - b} \right|} \right| + a{\left| {a + b - 2c - \left| {a - b} \right|} \right|^2} + b{\left| {a + b - 2c - \left| {a - b} \right|} \right|^2} + 2c{\left| {a + b - 2c - \left| {a - b} \right|} \right|^2} + 4d{\left| {a + b - 2c - \left| {a - b} \right|} \right|^2} + \left| {a - b} \right|{\left| {a + b - 2c - \left| {a - b} \right|} \right|^2} - 5{a^2}\left| {a + b - 2c + \left| {a - b} \right|} \right| - 10ab\left| {a + b - 2c + \left| {a - b} \right|} \right| - 5{b^2}\left| {a + b - 2c + \left| {a - b} \right|} \right| - 12ac\left| {a + b - 2c + \left| {a - b} \right|} \right| - 12bc\left| {a + b - 2c + \left| {a - b} \right|} \right| - 4{c^2}\left| {a + b - 2c + \left| {a - b} \right|} \right| - 16ad\left| {a + b - 2c + \left| {a - b} \right|} \right| - 16bd\left| {a + b - 2c + \left| {a - b} \right|} \right| + 16{d^2}\left| {a + b - 2c + \left| {a - b} \right|} \right| + 6a\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + 6b\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + 4c\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| - {\left| {a - b} \right|^2}\left| {a + b - 2c + \left| {a - b} \right|} \right| + 4a\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + 4b\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| - 8d\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + {\left| {a + b - 2c - \left| {a - b} \right|} \right|^2}\left| {a + b - 2c + \left| {a - b} \right|} \right| + a{\left| {a + b - 2c + \left| {a - b} \right|} \right|^2} + b{\left| {a + b - 2c + \left| {a - b} \right|} \right|^2} + 2c{\left| {a + b - 2c + \left| {a - b} \right|} \right|^2} + 4d{\left| {a + b - 2c + \left| {a - b} \right|} \right|^2} - \left| {a - b} \right|{\left| {a + b - 2c + \left| {a - b} \right|} \right|^2} - \left| {a + b - 2c - \left| {a - b} \right|} \right|{\left| {a + b - 2c + \left| {a - b} \right|} \right|^2} + 5{a^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 5{b^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 4{c^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 16{d^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 10ab\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 12ac\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 16ad\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 6a\left| {a - b} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + {\left| {a - b} \right|^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 12bc\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 16bd\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 6b\left| {a - b} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 4c\left| {a - b} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2a\left| {a + b - 2c - \left| {a - b} \right|} \right|4a\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 4b\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 8d\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 4c\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8d\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|{\left| {a + b - 2c + \left| {a - b} \right|} \right|^2}( - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|) + a{\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|^2} + b{\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|^2} + 2c{\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|^2} + 4d{\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|^2} + \left| {a - b} \right|{\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|^2} + \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| - 5{a^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 5{b^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 4{c^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 16{d^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 10ab\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 12ac\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 16ad\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 6a\left| {a - b} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + {\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|^2} - {\left| {a - b} \right|^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 12bc\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 16bd\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 6b\left| {a - b} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 4c\left| {a - b} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 4a\left| {a + b - 2c - \left| {a - b} \right|} \right|{\left| {a + b - 2c - \left| {a - b} \right|} \right|^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 4b\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 8d\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2a\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 4c\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 8d\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 2\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|{\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|^2} + 4a\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 4b\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 8d\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 2\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|\left| {\left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c - 4d} \right| + a{\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|^2} + b{\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|^2} + 2c{\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|^2} + 4d{\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|^2} - \left| {a - b} \right|{\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|^2} - \left| {a + b - 2c - \left| {a - b} \right|} \right|{\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|^2} - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|{\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right|^2} + \left. {\left. {2\sqrt {\frac{1}{4}{{(\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c - \left| {a - b} \right|} \right| + \left| {a - b} \right| - a - b - 2c - 4d)}^2}{{(\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a + b - 2c + \left| {a - b} \right|} \right| + 6a + 6b + 4c)}^2}{{(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)}^2} - 4096abcd( - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a - b} \right| + a + b + 2c + 4d)(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)} } \right)/16( - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a - b} \right| + a + b + 2c + 4d)(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)} \right)$$

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本帖最后由 isee 于 2019-7-11 14:58 编辑

回复 5# hbghlyj

超长公式,一般用 split 环境。

不过,在论坛,可以直接用\\强行拆分。

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又化简了一下,最终结果是$ - \left( {\left( { - 96abc - 32a{c^2} - 32b{c^2} + 32{c^3} - 160abd - 224acd - 224bcd - 32{c^2}d - 128a{d^2} - 128b{d^2} - 128c{d^2} + 128{d^3} + 8ac\left| {a + b - 2c - \left| {a - b} \right|} \right| + 8bc\left| {a + b - 2c - \left| {a - b} \right|} \right| - 16{c^2}\left| {a + b - 2c - \left| {a - b} \right|} \right| + 24ad\left| {a + b - 2c - \left| {a - b} \right|} \right| + 24bd\left| {a + b - 2c - \left| {a - b} \right|} \right| + 16cd\left| {a + b - 2c - \left| {a - b} \right|} \right| + 8c\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right| + 24d\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right| - 8ac\left| {a + b - 2c + \left| {a - b} \right|} \right| - 8bc\left| {a + b - 2c + \left| {a - b} \right|} \right| + 16{c^2}\left| {a + b - 2c + \left| {a - b} \right|} \right| - 24ad\left| {a + b - 2c + \left| {a - b} \right|} \right| - 24bd\left| {a + b - 2c + \left| {a - b} \right|} \right| - 16cd\left| {a + b - 2c + \left| {a - b} \right|} \right| + 8c\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + 24d\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| - 8c\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + } \right.} \right. - 8{c^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 32{d^2}\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 16ac\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 16bc\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 16cd\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8c\left| {a - b} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 4c\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8d\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right| + 8ab\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 24ad\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 24bd\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8d\left| {a - b} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2a\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 8d\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right|8{c^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 32{d^2}\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 4c\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 16ac\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 16bc\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 16cd\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 8c\left| {a - b} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2a\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 2\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - 8ab\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 24ad\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 24bd\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 8d\left| {a - b} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2a\left| {a + b - 2c - \left| {a - b} \right|} \right|2a\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 4c\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 8d\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 2\left| {a - b} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2b\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 4c\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2\left| {a - b} \right|\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {\left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c - 4d} \right| + 4a\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 4b\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 8d\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + 2\left| {a + b - 2c - \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| - 2\left| {a + b - 2c + \left| {a - b} \right|} \right|\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right|\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| \\± \left. {\left. {2\sqrt {\frac{1}{4}{{(\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c - \left| {a - b} \right|} \right| + \left| {a - b} \right| - a - b - 2c - 4d)}^2}{{(\left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a + b - 2c + \left| {a - b} \right|} \right| + 6a + 6b + 4c)}^2}{{(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)}^2} - 4096abcd( - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a - b} \right| + a + b + 2c + 4d)(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)} } \right)/16( - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a - b} \right| + a + b + 2c + 4d)(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)} \right)$我验证了10组实数,未发现问题,只是a,b,c,d都不能为0,否则分母为0.而且分母也很麻烦,整理后发现当且仅当a,b,c,d都为正数或都为负数时,分母为正数,即根号前面带负号,不然,如1,2,-1,-9,就带正号,所以只能写±号.这引发一个问题:四元情形能否像1#"提示"中二元和三元的情形那样,给出一个表达式,对任何实数都行得通?

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讨论分母也可以单独出一个题了:
设实数a,b,c,d,S=$16( - \left| {a + b + 2c - 4d - \left| {a - b} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right|} \right| - \left| {a + b - 2c - \left| {a - b} \right|} \right| - \left| {a - b} \right| + a + b + 2c + 4d)(\left| {a + b + 2c - 4d + \left| {a - b} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right|} \right| + \left| {a + b - 2c + \left| {a - b} \right|} \right| + \left| {a - b} \right| + a + b + 2c + 4d)$,求证:(1)S>0当且仅当a,b,c,d>0或a,b,c,d<0;(2)S=0当且仅当abcd=0.

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本帖最后由 hbghlyj 于 2019-7-12 21:35 编辑

再抛一砖,希望网友把这个表达式推广到n元,以及解决8#提出的问题.以下:
y=max(min(max(a,b),max(c,d)),min(a,b),min(c,d))
用绝对值函数最终表示为
y=$\frac{1}{8}( - \left| {a + b - c - d + \left| {a - b} \right| - \left| {c - d} \right|} \right| + \left| {a + b - c - d - \left| {a - b} \right| + \left| {c - d} \right|} \right| + \left| {2\left| {a - b} \right| + 2\left| {c - d} \right| - \left| {a + b - c - d + \left| {a - b} \right| - \left| {c - d} \right|} \right| - \left| {a + b - c - d - \left| {a - b} \right| + \left| {c - d} \right|} \right|} \right| + 2a + 2b + 2c + 2d)$
那么7#的问题解决了!
看来,殊途不一定同归,经验证,这个结果不仅对全体实数有效(分母不含变元),而且简短得多.

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