Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

Let $n$, $(n \geq 3)$ be a positive integer and the set $A$={$1,2,...,n$}. All the elements of $A$ are randomly arranged in a sequence $(a_1,a_2,...,a_n)$. The pair $(a_i,a_j)$ forms an $inversion$ if $1 \leq i \leq j \leq n$ and $a_i > a_j$. In how many different ways all the elements of the set $A$ can be arranged in a sequence that contains exactly $3$ inversions?

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

Let $ ABC$ be a triangle with sides $ BC\equal{}a, CA\equal{}b,AB\equal{}c$ and let $ D$ and $ E$ be the midpoints of $ AC$ and $ AB$, respectively. Prove that the medians $ BD$ and $ CE$ are perpendicular to each other if and only if $ b^2\plus{}c^2\equal{}5a^2$.

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.

In an exhibition where $2015$ paintings are shown, every participant picks a pair of paintings and writes it on the board. Then, Fake Artist (F.A.) chooses some of the pairs on the board, and marks one of the paintings in all of these pairs as "better". And then, Artist's Assistant (A.A.) comes and in his every move, he can mark $A$ better then $C$ in the pair $(A,C)$ on the board if for a painting $B$, $A$ is marked as better than $B$ and $B$ is marked as better than $C$ on the board. Find the minimum possible value of $k$ such that, for any pairs of paintings on the board, F.A can compare $k$ pairs of paintings making it possible for A.A to compare all of the remaining pairs of paintings. [b]P.S:[/b] A.A can decide $A_1>A_n$ if there is a sequence $ A_1 > A_2 > A_3 > \dots > A_{n-1} > A_n$ where $X>Y$ means painting $X$ is better than painting $Y$.

\[\int_0^1 \sqrt{x}\log(x)\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$. [i]Proposed by Aaron Lin[/i]

A [i]standard $n$-sided die[/i] has $n$ sides labeled $1$ to $n.$ Luis, Luke, and Sean play a game in which they roll a fair standard $4$-sided die, a fair standard $6$-sided die, and a fair standard $8$-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Compute the probability that they lose the game.

One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.

For which $k$ the number $N = 101 ... 0101$ with $k$ ones is a prime?

Several dwarves were lined up in a row, and then they lined up in a row in a different order. Is it possible that exactly one third of the dwarves have both of their neighbours remained and exactly one third of the dwarves have only one of their neighbours remained, if the number of the dwarves is a) 6; b) 9?

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

On the board is written an integer $N \geq 2$. Two players $A$ and $B$ play in turn, starting with $A$. Each player in turn replaces the existing number by the result of performing one of two operations: subtract 1 and divide by 2, provided that a positive integer is obtained. The player who reaches the number 1 wins. Determine the smallest even number $N$ requires you to play at least $2015$ times to win ($B$ shifts are not counted).

The points $A, B, C, D$ lie in this order on a circle with center $O$. Furthermore, the straight lines $AC$ and $BD$ should be perpendicular to each other. The base of the perpendicular from $O$ on $AB$ is $F$. Prove $CD = 2 OF$. [i](Karl Czakler)[/i]

Let $m \ge 2$ be an integer. Find the smallest integer $n>m$ such that for any partition of the set $\{m,m+1,\cdots,n\}$ into two subsets, at least one subset contains three numbers $a, b, c$ such that $c=a^{b}$.

Let $O_1O_2O_3$ be an acute angled triangle.Let $\omega_1$, $\omega_2$, $\omega_3$ be the circles with centres $O_1$, $O_2$, $O_3$ respectively such that any of two are tangent to each other. Circumcircle of $O_1O_2O_3$ intersects $\omega_1$ at $A_1$ and $B_1$, $\omega_2$ at $A_2$ and $B_2$, $\omega_3$ at $A_3$ and $B_3$ respectively. Prove that the incenter of triangle which can be constructed by lines $A_1B_1$, $A_2B_2$, $A_3B_3$ and the incenter of $O_1O_2O_3$ are coincide.

Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid [i]beautiful[/i] if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ [i]beautiful[/i] subgrids.

The graph of the equation $y = ax^2 + bx + c$ is shown in the diagram. Which of the following must be positive? [DIAGRAM NEEDED] $ \mathrm{(A) \ } a \qquad \mathrm{(B) \ } ab^2 \qquad \mathrm {(C) \ } b - c \qquad \mathrm{(D) \ } bc \qquad \mathrm{(E) \ } c - a$

A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Let $ABCDE$ be a convex pentagon with all five sides equal in length. The diagonals $AD$ and $EC$ meet in $S$ with $\angle ASE = 60^\circ$. Prove that $ABCDE$ has a pair of parallel sides.

The function $f$ is defined for non-negative integers and satisfies the condition $f(n) = f(f(n + 11))$, if $n \le 1999$ and $f(n) = n - 5$, if $n > 1999$. Find all solutions of the equation $f(n) = 1999$.

A train traveling from Aytown to Beetown meets with an accident after $ 1$ hr. It is stopped for $ \frac{1}{2}$ hr., after which it proceeds at four-fifths of its usual rate, arriving at Beetown $ 2$ hr. late. If the train had covered $ 80$ miles more before the accident, it would have been just $ 1$ hr. late. The usual rate of the train is: $ \textbf{(A)}\ \text{20 mph} \qquad \textbf{(B)}\ \text{30 mph} \qquad \textbf{(C)}\ \text{40 mph} \qquad \textbf{(D)}\ \text{50 mph} \qquad \textbf{(E)}\ \text{60 mph}$

Five points $A, B, C, D$, and $E$ in three-dimensional Euclidean space have the property that $AB = BC = CD = DE = EA = 1$ and $\angle ABC = \angle BCD =\angle CDE = \angle DEA = 90^o$ . Find all possible $\cos(\angle EAB)$.