Let $({{A}_{n}})_{n=1}^{\infty }$ and $({{a}_{n}})_{n=1}^{\infty }$ be sequences of positive integers. Assume that for each positive integer $x$, there is a unique positive integer $N$ and a unique $N-tuple$ $({{x}_{1}},...,{{x}_{N}})$ such that $0\le {{x}_{k}}\le {{a}_{k}}$ for $k=1,2,...N$, ${{x}_{N}}\ne 0$, and $x=\sum\limits_{k=1}^{N}{{{A}_{k}}{{x}_{k}}}$. (a) Prove that ${{A}_{k}}=1$ for some $k$; (b) Prove that ${{A}_{k}}={{A}_{j}}\Leftrightarrow k=j$; (c) Prove that if ${{A}_{k}}\le {{A}_{j}}$, then $\left. {{A}_{k}} \right|{{A}_{j}}$.

Let $ABCD$ be a square of side length 2, and let $M$ and $N$ be points on the sides $AB$ and $CD$ respectively. The lines $CM$ and $BN$ meet at $P$, while the lines $AN$ and $DM$ meet at $Q$. Prove that $\left| PQ \right|\ge 1$.

Let $n$ integers on the real axis be colored. Determine for which positive integers $k$ there exists a family $K$ of closed intervals with the following properties: i) The union of the intervals in $K$ contains all of the colored points; ii) Any two distinct intervals in $K$ are disjoint; iii) For each interval $I$ at $K$ we have ${{a}_{I}}=k.{{b}_{I}}$, where ${{a}_{I}}$ denotes the number of integers in $I$, and ${{b}_{I}}$ the number of colored integers in $I$.

A circle is tangent to sides $AD,\text{ }DC,\text{ }CB$ of a convex quadrilateral $ABCD$ at $\text{K},\text{ L},\text{ M}$ respectively. A line $l$, passing through $L$ and parallel to $AD$, meets $KM$ at $N$ and $KC$ at $P$. Prove that $PL=PN$.

Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.

Show that there is no function $f:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}$ such that $f(x+y)>f(x)(1+yf(x))$ for all $x,y\in {{\mathbb{R}}^{+}}$.