Školjka

%V0 Let $n$ be a positive integer. Show that the numbers $$\binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}$$ are congruent modulo $2^n$ to $1$, $3$, $5$, $\ldots$, $2^n - 1$ in some order. Proposed by Duskan Dukic, Serbia

%V0 For every integer $k \geq 2,$ prove that $2^{3k}$ divides the number $$\binom{2^{k + 1}}{2^{k}} - \binom{2^{k}}{2^{k - 1}}$$ but $2^{3k + 1}$ does not. Author: unknown author, Poland

%V0 Find all positive integers $n$ such that there exists a unique integer $a$ such that $0\leq a < n!$ with the following property: $$n!\mid a^n + 1$$

%V0 Let $a$, $b$, $c$, $d$, $e$, $f$ be positive integers and let $S = a+b+c+d+e+f$. Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$. Prove that $S$ is composite.

%V0 Let $b$ be an integer greater than $5$. For each positive integer $n$, consider the number $$x_n = \underbrace{11\cdots1}_{n - 1}\underbrace{22\cdots2}_{n}5,$$ written in base $b$. Prove that the following condition holds if and only if $b = 10$: there exists a positive integer $M$ such that for any integer $n$ greater than $M$, the number $x_n$ is a perfect square.

%V0 Is there a positive integer $m$ such that the equation $${1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c}$$ has infinitely many solutions in positive integers $a,b,c$?