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}.

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