Vietnam Math Out Standing Student 2012-13

Problem 01: Solve the equation of: `x^(4n)`+`\sqrt{x^{2n}+2012}``=2012`    for all `n\in N`.

Problem 02: It is given `(U_n)` which is determined by: 

                    `U_1=3`

                    `U_(n+1)=1/3(2U_n+3/U_n^2)`;     for all `n\in N`.

                    Let's finding the limit of: `\lim_{n\rightarrow\infty}(u_n)`.

Problem 03: It is given three non-negative real numbers `x,y,z` , Prove that:

                    `1/x+1/y+1/z``>36/(9+x^2y^2+y^2^2+z^2x^2)`.

Problem 04: Find roots positive numbers of equation:

                    `\sqrt{x+2\sqrt3}=\sqrt y+\sqrt z`

Solution

Problem 01: We had the equation: `x^(4n)`+`\sqrt{x^{2n}+2012}``=2012`    for all `n\in N`.

                    Let's `t=x^(2n)` and replace into our equation above:

                    `t^2+\sqrt{t+2012}``=2012`

                      `t^2+t+1/4=t+2012-\sqrt{t+2012}+1/4`

                      `(t+1/2)^2=(\sqrt{t+2012}-1/2)^2`

                     `t+1=\sqrt{t+2012}`

                       `t^2+t-2011=0`         `(1)`

Solve the equation `(1)` we got: `t=(-1+\sqrt{8045})/2`

Then, we replace the value of `t` in to`t=x^(2n)` we will get the roots of our problem.