Školjka

%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 Let $a_0$, $a_1$, $a_2$, $\ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $\gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $a_n\ge 2^n$ for all $n\ge 0$. Proposed by Morteza Saghafian, Iran

%V0 Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

%V0 Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.

%V0 Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write $$\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots ,$$ where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by $$f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}$$ a) Prove that $S$ is infinite. b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$

%V0 Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.