Olympiad problems

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Found problems: 3

1997 Akdeniz University MO, 2

Tags: easy inequality , inequalities

If $x$ and $y$ are positive reals, prove that $$x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2$$

Tags: easy inequality , inequalities

Let $x,y,z$ be non-negative numbers such that $x+y+z \leq 3$. Prove that $$\frac{2}{1+x}+\frac{2}{1+y}+\frac{2}{1+z} \geq 3$$

Tags: inequalities , easy inequality , easy

Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that $$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$