Školjka

%V0 Let $k$ be a positive integer. Show that if there exists a sequence $a_0$, $a_1$, ... of integers satisfying the condition $$a_n=\frac{a_{n-1}+n^k}{n} \text{,} \qquad \forall n \geqslant 1 \text{,}$$ then $k-2$ is divisible by $3$. Proposed by Turkey

%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 We define a sequence $\left(a_{1},a_{2},a_{3},...\right)$ by setting $$a_{n} = \frac {1}{n}\left(\left[\frac {n}{1}\right] + \left[\frac {n}{2}\right] + \cdots + \left[\frac {n}{n}\right]\right)$$ for every positive integer $n$. Hereby, for every real $x$, we denote by $\left[x\right]$ the integral part of $x$ (this is the greatest integer which is $\leq x$). a) Prove that there is an infinite number of positive integers $n$ such that $a_{n + 1} > a_{n}$. b) Prove that there is an infinite number of positive integers $n$ such that $a_{n + 1} < a_{n}$.

%V0 A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.

%V0 Define the sequences $a_n, b_n, c_n$ as follows. $a_0 = k, b_0 = 4, c_0 = 1$. If $a_n$ is even then $a_{n + 1} = \frac {a_n}{2}$, $b_{n + 1} = 2b_n$, $c_{n + 1} = c_n$. If $a_n$ is odd, then $a_{n + 1} = a_n - \frac {b_n}{2} - c_n$, $b_{n + 1} = b_n$, $c_{n + 1} = b_n + c_n$. Find the number of positive integers $k < 1995$ such that some $a_n = 0$.