Školjka

%V0 Define a sequence $<f(n)>^{\infty}_{n=1}$ of positive integers by $$f(1) = 1$$ and $f(n) =\left\{\begin{array}{ll}f(n-1)-n &\text{ if }f(n-1) > n;\\ f(n-1)+n &\text{ if }f(n-1)\leq n,\end{array}\right$ for $n \geq 2.$ Let $S = \{n \in \mathbb{N} | f(n) = 1993\}.$ (i) Prove that $S$ is an infinite set. (ii) Find the least positive integer in $S.$ (iii) If all the elements of $S$ are written in ascending order as $$n_1 < n_2 < n_3 < \ldots ,$$ show that $$\lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3.$$

%V0 Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$

%V0 Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define $$f(s) = (n_0, m + n - n_0).$$ Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that $$f^t(s) = s,$$ where $$f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s).$$ If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$

%V0 Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that (i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$.

%V0 Consider two monotonically decreasing sequences $\left( a_k\right)$ and $\left( b_k\right)$, where $k \geq 1$, and $a_k$ and $b_k$ are positive real numbers for every k. Now, define the sequences $c_k = \min \left( a_k, b_k \right)$; $A_k = a_1 + a_2 + ... + a_k$; $B_k = b_1 + b_2 + ... + b_k$; $C_k = c_1 + c_2 + ... + c_k$ for all natural numbers k. (a) Do there exist two monotonically decreasing sequences $\left( a_k\right)$ and $\left( b_k\right)$ of positive real numbers such that the sequences $\left( A_k\right)$ and $\left( B_k\right)$ are not bounded, while the sequence $\left( C_k\right)$ is bounded? (b) Does the answer to problem (a) change if we stipulate that the sequence $\left( b_k\right)$ must be $\displaystyle b_k = \frac {1}{k}$ for all k ?

%V0 Each positive integer $a$ is subjected to the following procedure, yielding the number $d = d\left(a\right)$: (a) The last digit of $a$ is moved to the first position. The resulting number is called $b$. (b) The number $b$ is squared. The resulting number is called $c$. (c) The first digit of $c$ is moved to the last position. The resulting number is called $d$. (All numbers are considered in the decimal system.) For instance, $a = 2003$ gives $b = 3200$, $c = 10240000$ and $d = 02400001 = 2400001 = d\left(2003\right)$. Find all integers a such that $d\left( a\right) =a^2$.