ptt-math [分析] 求範圍

文章:[分析] 求範圍 by jazzter@ptt

題目:證明下列不等式成立。

$$ \frac{2(a_1^2+a_2^2+a_3^2+a_4^2)}{(a_1+a_2+a_3+a_4)^2} - \frac{(a_1^3+a_2^3+a_3^3+a_4^3)}{(a_1+a_2+a_3+a_4)^3} \geq \frac{7}{16} $$

解答:直接暴力通分,然後交叉相乘。然後可以化簡拆開成下列諸項:

$$ \begin{aligned} &3(a_1^3 + a_2^2a_3 + a_2a_3^2 - 3a_1a_2a_3)\\ &3(a_1^3 + a_2^2a_4 + a_2a_4^2 - 3a_1a_2a_4)\\ &3(a_1^3 + a_3^2a_4 + a_3a_4^2 - 3a_1a_3a_4)\\ &3(a_2^3 + a_1^2a_3 + a_1a_3^2 - 3a_1a_2a_3)\\ &3(a_2^3 + a_1^2a_4 + a_1a_4^2 - 3a_1a_2a_4)\\ &3(a_2^3 + a_3^2a_4 + a_3a_4^2 - 3a_2a_3a_4)\\ &3(a_3^3 + a_1^2a_2 + a_1a_2^2 - 3a_1a_2a_3)\\ &3(a_3^3 + a_1^2a_4 + a_1a_4^2 - 3a_1a_3a_4)\\ &3(a_3^3 + a_2^2a_4 + a_2a_4^2 - 3a_2a_3a_4)\\ &3(a_4^3 + a_1^2a_2 + a_1a_2^2 - 3a_1a_2a_4)\\ &3(a_4^3 + a_1^2a_3 + a_1a_3^2 - 3a_1a_3a_4)\\ &3(a_4^3 + a_2^2a_3 + a_2a_3^2 - 3a_2a_3a_4)\\ &5(a_1^2a_2 + a_2^2a_3 + a_3^2a_1 - 3a_1a_2a_3)\\ &5(a_1^2a_2 + a_2^2a_4 + a_4^2a_1 - 3a_1a_2a_4)\\ &5(a_1^2a_3 + a_3^2a_4 + a_4^2a_1 - 3a_1a_3a_4)\\ &5(a_2^2a_3 + a_3^2a_4 + a_4^2a_2 - 3a_2a_3a_4) \end{aligned} $$

由算幾不等式可得每一項皆是正的,故本題得證。

其它強者的解法:

updatedupdated2021-04-032021-04-03