IMO training

Given that $a,b,c> 0$ and $a+b+c=ab+bc+ca$.Prove that: \[\sum \frac{a^2}{a^2-a+1}\leq 3\]
Let $x;y;z >0$ . Prove that $xyz+2(x^2+y^2+z^2)+8\geq 5(x+y+z)$
Revised version,find all the constant $k$ such that $xyz+k(x^2+y^2+z^2)+8\geq (k+3)(x+y+z)$ for all $x;y;z >0$
Let $x;y;z >0$ Prove that $(x+y+z)^{2}(x^2+y^2)(y^2+z^2)(z^2+x^2)\geq 8(x^2y^2+y^2z^2+z^2x^2)^{2}$
Let $0\leq a;b;c\leq 1$ . Put $x=1-a+ab;y=1-b+bc;z=1-c+ca$
a) Prove that $x+y+z\geq 2$
b)Prove that $x^2+y^2+z^2\geq \frac{3}{2}$
c)Prove or disprove $x^3+y^3+z^3\geq \frac{5}{4}$
Let be given positive integer $n$ . Find the least real number $k$ such that $(xy)^{k}(x^n+y^n)\leq 2$ for all positive real numbers $x;y$ satisfying the condition $x+y=2$

These problems has been taken from Vietnam TST practice for Hong Kong IMO 2016.

#Algebra

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`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$