Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information: $i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$ $ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen: a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows: $a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$ b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$. The sequence $C_0$ that appears in the screen is the following: $a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$ Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.

$N$ lines lie on a plane, no two of which are parallel and no three of which are concurrent. Prove that there exists a non-self-intersecting broken line $A_0A_1A_2A_3...A_N$ with $N$ parts, such that on each of the $N$ lines lies exactly one of the $N$ segments of the line.

A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible? $ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$

Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square. [i]David Yang, Alex Zhu.[/i]

Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?

A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

Let $n>2$ be an integer. In a country there are $n$ cities and every two of them are connected by a direct road. Each road is assigned an integer from the set $\{1, 2,\ldots ,m\}$ (different roads may be assigned the same number). The [i]priority[/i] of a city is the sum of the numbers assigned to roads which lead to it. Find the smallest $m$ for which it is possible that all cities have a different priority.