Olympiad problems

the 11th XMO, 2

Suppose $a,b,c>0$ and $abc=64$, show that $$\sum_{cyc}\frac{a^2}{\sqrt{a^3+8}\sqrt{b^3+8}}\ge\frac{2}{3}$$

inequalities algebraic inequality

2017 Pan-African Shortlist, I?

Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$ In which cases do we have equality?

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2018 Turkey Junior National Olympiad, 4

For all $x,y,z$ positive real numbers, find the all $c$ positive real numbers that providing $$\frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2$$

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2015 Irish Math Olympiad, 6

Suppose $x,y$ are nonnegative real numbers such that $x + y \le 1$. Prove that $8xy \le 5x(1 - x) + 5y(1 - y)$ and determine the cases of equality.

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2015 Saudi Arabia JBMO TST, 1

Let $a,b,c$ be positive real numbers. Prove that: $\left (a+b+c \right )\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right ) \geq 9+3\sqrt[3]{\frac{(a-b)^2(b-c)^2(c-a)^2}{a^2b^2c^2}}$

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2017 Pan African, Problem 2

Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$ In which cases do we have equality?

inequalities algebraic inequality AM-GM inequality Cauchy-Schwarz inequality algebra