Školjka

%V0 $(USS 2)$ Prove that for $a > b^2$, the identity $$\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b$$ holds.

%V0 Let $x, y, z$ be real numbers each of whose absolute value is different from $\displaystyle \frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that $$\frac{3x-x^{3}}{1-3x^{2}}+\frac{3y-y^{3}}{1-3y^{2}}+\frac{3z-z^{3}}{1-3z^{2}}=\frac{3x-x^{3}}{1-3x^{2}}\cdot\frac{3y-y^{3}}{1-3y^{2}}\cdot\frac{3z-z^{3}}{1-3z^{2}}$$

%V0 Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $\displaystyle P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$

%V0 Let $n \in \mathbb{Z}^+$ and let $a, b \in \mathbb{R}.$ Determine the range of $x_0$ for which $$\sum^n_{i=0} x_i = a \text{ and } \sum^n_{i=0} x^2_i = b,$$ where $x_0, x_1, \ldots , x_n$ are real variables.

%V0 Let $x$, $y$, $z$ be real numbers satisfying $x^2+y^2+z^2+9=4(x+y+z)$. Prove that $$x^4+y^4+z^4+16(x^2+y^2+z^2) \ge 8(x^3+y^3+z^3)+27$$ and determine when equality holds.

%V0 Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that $$\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.$$