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}. Continue reading “Analyzing Squares (1)”