# Thread: Inequality problem

1. ## Inequality problem

If $\displaystyle a,b > 0$ and $\displaystyle a+b=1$
prove that

$\displaystyle (a+\frac{1}{a})^2+(b+\frac{1}{b})^2 \ge \frac{25}{2}$

Where I can find more problems like that ?

2. Originally Posted by gilyos
If $\displaystyle a,b > 0$ and $\displaystyle a+b=1$
prove that

$\displaystyle (a+\frac{1}{a})^2+(b+\frac{1}{b})^2 \ge \frac{25}{2}$

Where I can find more problems like that ?
The function $\displaystyle f(x)=\left(x+\tfrac{1}{x}\right)^2$ is convex. That pretty much finishes it. Look up IMO.

3. Imo ?!

4. Originally Posted by gilyos
Imo ?!
International Math Olympiad.

5. in the IMO the problems are difficult ... i need something like this problem
and if u can explain who I prove this problem

6. Originally Posted by gilyos
in the IMO the problems are difficult ... i need something like this problem
and if u can explain who I prove this problem
Oh...I'm sorry that you don't want hard problems. Do you know what convexity is? Is this for a class? I mean, we need a little more.

7. Originally Posted by gilyos
and if u can explain who I prove this problem
Do you know Jensen's Inequality? I think that is what Drexel28 had in mind.

8. Where I can find inequalites problems with Cauchy , Bernoulli's inequality , Inequality of arithmetic and geometric means ??? Book or site or something else ...

9. See this book for example, it's excellent.
Here you have Olympiad Inequalities by Thomas Mildford, and here : Topics in Inequalities by Hojoo Lee.
There are several pages with inequality problems, just google a bit.

Have fun