# IMO training

1. Given that $$a,b,c> 0$$ and $$a+b+c=ab+bc+ca$$.Prove that: $\sum \frac{a^2}{a^2-a+1}\leq 3$

2. 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 $$x;y;z >0$$

3. Let $$x;y;z >0$$ Prove that $$(x+y+z)^{2}(x^2+y^2)(y^2+z^2)(z^2+x^2)\geq 8(x^2y^2+y^2z^2+z^2x^2)^{2}$$

4. 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 $x^3+y^3+z^3\geq \frac{5}{4}$

5. 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.

Note by Ms Ht
2 years ago

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