The Fibonacci sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\in\mathbb N$. (a) Show that $f_{1005}$ is divisible by $10$. (b) Show that $f_{1005}$ is not divisible by $100$.

Let $n \ge k \ge 3$ be integers. Show that for every integer sequence $1 \le a_1 < a_2 < . . . < a_k \le n$ one can choose non-negative integers $b_1, b_2, . . . , b_k$, satisfying the following conditions: [list=i] [*] $0 \le b_i \le n$ for each $1 \le i \le k$, [*] all the positive $b_i$ are distinct, [*] the sums $a_i + b_i$, $1 \le i \le k$, form a permutation of the first $k$ terms of a non-constant arithmetic progression. [/list]

A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$. Does there exist a sum-full zero-sum-free set of integers? [i]Romania (Dan Schwarz)[/i]

Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$. Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$. Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.

For an integer $n\geq 1$ let $a(n)$ denote the total number of carries which arise when adding $2017$ and $n\cdot 2017$. The first few values are given by $a(1)=1$, $a(2)=1$, $a(3)=0$, which can be seen from the following: \begin{align*} 001 &&001 && 000 \\ 2017 &&4034 &&6051 \\ +2017 &&+2017 &&+2017\\ =4034 &&=6051 &&=8068\\ \end{align*} Prove that $$a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.$$

Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.

Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.

We denote by $S(k)$ the sum of digits of a positive integer number $k$. We say that the positive integer $a$ is $n$-good, if there is a sequence of positive integers $a_0$, $a_1, \dots , a_n$, so that $a_n = a$ and $a_{i + 1} = a_i -S (a_i)$ for all $i = 0, 1,. . . , n-1$. Is it true that for any positive integer $n$ there exists a positive integer $b$, which is $n$-good, but not $(n + 1)$-good? A. Antropov

Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

An $n\times n$ chessboard is given, where $n$ is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is [i]alternative[/i] if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board [i]nonalternative[/i]. For what values of $n$ do we always get from the $n\times n$ chessboard an alternative board?\\ \\ [i](Alexandru Petrescu and Andra Elena Mircea)[/i]

Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively. a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint? b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint? Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.

Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following. For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$. Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.

Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$

A positive integer is called [i]fancy[/i] if it can be expressed in the form $$2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}},$$ where $a_1,a_2, \cdots, a_{100}$ are non-negative integers that are not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is a [i]fancy[/i] number. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

Find all functions $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$, for which $f(k+1)>f(f(k)) \quad \forall k \geq 1$.