Record of some solved inequalities

可嘗試以下問題 (1, 2, 4, 6, 7, 8 都是 AL 知識範圍內)，請不要使用暴力的方法 (暴力通分) 解決問題。

Problem 1. Let $  a,b,c>0$ and $  abc=1$. Prove that  \[\frac{a}{a+b+1}+\frac{b}{b+c+1}+\frac{c}{c+a+1}\geq 1.\]
Problem 2. Let $ x,y,z>0$; $  x+y+z=1$ prove that \[\sqrt{\frac{x}{yz}}+\sqrt{\frac{y}{zx}}+\sqrt{\frac{z}{xy}}\ge 2\left(\sqrt{\frac{x}{(x+y)(x+z)}}+\sqrt{\frac{y}{(y+z)(y+x)}}+\sqrt{\frac{z}{(z+x)(z+y)}}\right).\]
以下雖然不難，卻十分漂亮，值得牢記！經驗告訊我 $  ab+bc+ca$ 和 $  a+b+c$ 也是十分常見的因子。

Problem 3. Prove that for any $a,b,c\ge 0$, we always have \[
9(a+b)(b+c)(c+a)\ge 8(a+b+c)(ab+bc+ca)
\] and \[ (a+b+c)(a^{2}+b^{2}+c^{2})+9abc\ge 2(a+b+c)(ab+bc+ca).
\]
Problem 4. When $  a+b+c=3$, $  a,b,c\ge 0$, prove that \[\frac{a+3}{3a+bc}+\frac{b+3}{3b+ca}+\frac{c+3}{3c+ab}\ge 3.\]
Problem 5. Let $ a,b,c>0$, show that \[a^{2}+b^{2}+c^{2}+2abc+1\ge 2(ab+bc+ac).\]
Problem 6. Let $  x,y,z>0$, prove that \[\frac{xy}{x^{2}+y^{2}+2z^{2}}+\frac{yz}{y^{2}+z^{2}+2x^{2}}+\frac{zx}{z^{2}+x^{2}+2y^{2}}\leq\frac{3}{4}.\]
Problem 7. Let $  a,b,c$ be positive real numbers such that $  abc=1$. Prove that \[\frac{1}{a+b^{2}+c^{3}}+\frac{1}{b+c^{2}+a^{3}}+\frac{1}{c+a^{2}+b^{3}}\leq 1.\]
Problem 8. $  a,b,c$ are real positive numbers, prove that \[\frac{ab}{c(c+a)}+\frac{bc}{a(a+b)}+\frac{ca}{b(b+c)} \geq \frac{a}{c+a}+\frac{b}{a+b}+\frac{c}{b+c}.\]