Školjka

%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 Is it possible to find 100 positive integers not exceeding 25,000, such that all pairwise sums of them are different?

%V0 Let $a_1 = 11^{11}, \, a_2 = 12^{12}, \, a_3 = 13^{13}$, and $a_n = |a_{n - 1} - a_{n - 2}| + |a_{n - 2} - a_{n - 3}|, n \geq 4.$ Determine $a_{14^{14}}$.

%V0 Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

%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 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 $P.$