×

# 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
1 year, 6 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$