Školjka

%V0 For what real values of $x$ is $$\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A$$ given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

%V0 Given any integer $n \geq 2,$ assume that the integers $a_1, a_2, \ldots, a_n$ are not divisible by $n$ and, moreover, that $n$ does not divide $\sum^n_{i=1} a_i.$ Prove that there exist at least $n$ different sequences $(e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $\sum^n_{i=1} e_i \cdot a_i$ is divisible by $n.$

%V0 An infinite sequence $\,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be bounded if there is a constant $\,C\,$ such that $\, \vert x_{i} \vert \leq C\,$ for every $\,i\geq 0$. Given any real number $\,a > 1,\,$ construct a bounded infinite sequence $x_{0},x_{1},x_{2},\ldots \,$ such that $$\vert x_{i} - x_{j} \vert \vert i - j \vert^{a}\geq 1$$ for every pair of distinct nonnegative integers $i, j$.

%V0 For any positive integer $x$ define $g(x)$ as greatest odd divisor of $x,$ and $$f(x) =\begin{cases}\frac{x}{2}+\frac{x}{g(x)}&\text{if\ \(x\) is even},\\ 2^{\frac{x+1}{2}}&\text{if\ \(x\) is odd}.\end{cases}$$ Construct the sequence $x_1 = 1, x_{n + 1} = f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $n$ such that $x_n = 1992,$ and determine whether $n$ is unique.

%V0 Let $R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $R_1 = (1),$ and if $R_{n - 1} = (x_1, \ldots, x_s),$ then $$R_n = (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).$$ For example, $R_2 = (1, 2),$ $R_3 = (1, 1, 2, 3),$ $R_4 = (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $n > 1,$ then the $k$th term from the left in $R_n$ is equal to 1 if and only if the $k$th term from the right in $R_n$ is different from 1.

%V0 Find all pairs $(a,b)$ of positive integers that satisfy the equation: $a^{b^2} = b^a$.