Školjka

%V0 A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

%V0 Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1|a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

%V0 Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor$ for all positive integers $n$. (Note that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)

%V0 A sequence of integers $a_{1},a_{2},a_{3},\ldots$ is defined as follows: $a_{1} = 1$ and for $n\geq 1$, $a_{n + 1}$ is the smallest integer greater than $a_{n}$ such that $a_{i} + a_{j}\neq 3a_{k}$ for any $i,j$ and $k$ in $\{1,2,3,\ldots ,n + 1\}$, not necessarily distinct. Determine $a_{1998}$.

%V0 Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

%V0 Niz $\{ a_n \} $ je zadan na ovaj način: $$ a_0=0, \ a_1=1, \ a_n=2a_{n-1}+a_{n-2}, \ n>1.$$ Dokažite da $2^k$ dijeli $a_n$ ako i samo ako $2^k$ dijeli $n$.