I was said that a geometrical method is very useful to solve problems. Do you have any idea? Any book can you suggest me to read?
Thanks.
Well, if "geometrical" includes trigonometry and vectors I have some problems that can be solved easier like that. Also geometrical method is useful in problems with complex numbers (and, yes, I have problems that can be solved easier that way too).
Books - um, a good and complet one that I know is "Complex numbers from A to Z" - Titu Andreescu, Dorin Andrica.
$\displaystyle \frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+...+\frac{a_n^ 2}{b_n}\geq \frac{(a_1+a_2+...+a_n)^2}{b_1+b_+...+b_n},$
Equality if and only if $\displaystyle \frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n }$
Demonstration:
$\displaystyle (x_1^2+x_2^2+...+x_n^2)(y_1^2+y_2^2+...+y_n^2)\geq (x_1y_1+x_2y_2+...+x_ny_n)^2$ (Cauchy's inequality)
Let $\displaystyle x_i=\frac{a_i}{\sqrt{b_i}}$ and $\displaystyle y_i=\sqrt{b_i}$, then:
$\displaystyle \sum_{i=1}^{n}\frac{a_i^2}{{b_i}}\cdot \sum_{i=1}^{n}b_i\geq \left ( \sum_{i=1}^{n}a_i^2 \right )\Rightarrow$ $\displaystyle \sum_{i=1}^{n}\frac{a_i^2}{{b_i}}\geq \frac{\left ( \sum_{i=1}^{n}a_i^2 \right )}{\sum_{i=1}^{n}b_i}$ which is exactly Titu's inequality.