Školjka

%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 Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.

%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 Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: $x_i= 2^i$ if $0 \leq i\leq m-1$ and $x_i = \sum_{j=1}^{m}x_{i-j},$ if $i\geq m$. Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$.

%V0 Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$, ..., $a_n$ satisfying $$a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1$$ for every $k$ with $2 \leqslant k \leqslant n-1$. Proposed by North Korea

%V0 Neka je skup prirodnih brojeva podijeljen u intervale na sljedeći način: U prvom intervalu je broj 1, u drugom brojevi 2 i 3, u trećem 4, 5 i 6 i u svakom idućem jedan broj više nego u prethodnom (brojevi u intervalima su uzastopni). Neka je $p_i$ udio prostih brojeva u $i$-tom intervalu. a) Dokaži ili opovrgni: Postoji beskonačno brojeva $k$ za koje je $ p_{k+1} < p_k$. b) Dokaži ili opovrgni: Postoji beskonačno brojeva $k$ za koje je $ p_{k+1} > p_k$.