內地的高考

不等式是內地高考一個重要的課題。懷着好奇的心去做做看，不懂的還蠻多的...。以下都是些解了的題目，在此不提供解答了。

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Problem 1. Prove that for any $a,b,c\in\mathbb{R}$, $\displaystyle \sqrt{a^2+(1-b)^2}+\sqrt{b^2+(1-c)^2}+\sqrt{c^2+(1-a)^2}\ge \frac{3\sqrt{2}}{2}.$

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Problem 2 (Mircea Lascu).  Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $\displaystyle \frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{2}}\ge \sqrt{a}+\sqrt{b}+\sqrt{c}+3.$

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Problem 3. Let $a,b,c,x,y,z$ be positive real numbers such that $x+y+z=1$. Prove that $\displaystyle ax+by+cz+2\sqrt{(xy+yz+zx)(ab+bc+ca)}\leq a+b+c.$