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题目:求下列方程的实数解:$x^5+(x+1)^5+(x+2)^5+(x+3)^5=0$. 解:原方程可化为 $[x+(x+3)][(x+3)^4-(x+3)^3x+(x+3)^2x^2-(x+3)x^3+x^4]$ $+[(x+1)+(x+2)][(x+2)^4-(x+2)^3(x+1)+(x+2 阅读全文
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题目:已知$a,b,c\in R$,求证:$(1+a^2)(1+b^2)(1+c^2)(1+a^2b^2c^2)\geq (1+abc(a+b+c))^2$. 证明:因为$(1+a^2)(1+b^2)(1+c^2)(1+a^2b^2c^2)-(1+abc(a+b+c))^2$ $=\frac{1}{ 阅读全文
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题目:设正数$a_{1},a_{2},\cdots,a_{n}$满足$a_{1}+a_{2}+\cdots+a_{n}=1$,求证: $\left(\frac{1}{a_{1}^{3}}-1\right)\left(\frac{1}{a_{2}^{3}}-1\right)\cdots\left(\f 阅读全文