Školjka

%V0 Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

%V0 Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that $$(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)$$ is divisible by the product $c_1c_2\ldots c_n$.

%V0 If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $\displaystyle x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$

%V0 Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$

%V0 Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: $$\min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}.$$

%V0 A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$