# Analyzing Squares (1)

Problem 1. Let $a,b,c$ be positive real numbers. Prove that $\displaystyle\frac{a^3}{a^2+2b^2}+\frac{b^3}{b^2+2c^2}+\frac{c^3}{c^2+2a^2}\geq \frac{a^3}{2a^2+b^2}+\frac{b^3}{2b^2+c^2}+\frac{c^3}{2c^2+a^2}.$

Problem 2. Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2=1$. Prove that $\displaystyle a+b+c+\frac{1}{abc}\geq 4\sqrt{3}.$

Problem 3. Let $a,b,c$ be non-negative real nunbers. Prove that $\displaystyle a^3+b^3+c^3+3abc\geq ab\sqrt{2a^2+2b^2}+bc\sqrt{2b^2+2c^2}+ca\sqrt{2c^2+2a^2}.$

Problem 4. Let $a,b,c$ be positive real nunbers such that $abc=1$. Prove that $\displaystyle \frac{1}{(1+a)^3}+\frac{1}{(1+b)^3}+\frac{1}{(1+c)^3}+\frac{5}{(1+a)(1+b)(1+c)}\geq 1.$

Problem 5. Let $a,b,c$ be real numbers such that $a,b,c\geq 1$ and $a+b+c=9$. Prove that $\sqrt{ab+bc+ca}\leq\sqrt{a}+\sqrt{b}+\sqrt{c}.$

Problem 6. Let $a,b,c$ be positive real nunbers such that $ab+bc+ca=1$. Prove that $\displaystyle \frac{1+a^2b^2}{(a+b)^2}+\frac{1+b^2c^2}{(b+c)^2}+\frac{1+c^2a^2}{(c+a)^2}\geq\frac{5}{2}.$

Problem 7. Let $a,b,c$ be positive real nunbers such that $a+b+c=1$. Prove that $\displaystyle \frac{\sqrt{a}}{b+ca}+\frac{\sqrt{b}}{c+ab}+\frac{\sqrt{c}}{a+bc}\geq\frac{9\sqrt{3}}{4}.$

Problem 8. Let $a,b,c$ be positive real numbers. Prove that $\displaystyle a^2+b^2+c^2\geq\frac{9ab^3}{5a^2+4b^2}+\frac{9bc^3}{5b^2+4c^2}+\frac{9ca^3}{5c^2+4a^2}.$