Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)

Consider a sequence $F_0=2$, $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$. If each term of the sequence can be written in the form $a\cdot r_1^n+b\cdot r_2^n$, what is the positive difference between $r_1$ and $r_2$?

The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$. [i]Remark:[/i] A sum over the elements of the empty set is assumed to be $0$.

Find the smallest possible value of the nonnegative number $\lambda$ such that the inequality $$\frac{a+b}{2}\geq\lambda \sqrt{ab}+(1-\lambda )\sqrt{\frac{a^2+b^2}{2}}$$ holds for all positive real numbers $a, b$.

There are $n{}$ currencies in a country, numbered from 1 to $n{}.$ In each currency, only non-negative integers are possible amounts of money. A person can have only one currency at any time. A person can exchange all the money he has from currency $i{}$ to currency $j{}$ at the rate of $\alpha_{ij}$ which is a positive real number. If he had $d{}$ units of currency $i{}$ he instead receives $\alpha_{ij}d$ units of currency $j{}$ while this number is rounded to the nearest integer; a number of the form $t-1/2$ is rounded to $t{}$ for any integer $t{}.$ It is known that $\alpha_{ij}\alpha_{jk}=\alpha_{ik}$ and $\alpha_{ii}=1$ for every $i,j,k.$ Can there be a person who can get rich indefinitely? [i]Proposed by I. Bogdanov[/i]

Let $x_1 \le x_2 \le \dots < x_n$ (with $n \ge 2$) and let $S$ be the set of all the $x_i$. Let $T$ be a randomly chosen subset of $S$. What is the expected value of the indexed alternating sum of $T$ ? Express your answer in terms of the $x_i$. Note: We define the indexed alternating sum of $T$ as \[ \sum_{i=1}^{|T|} (-1)^{i+1}(i) T[i], \] where $T[i]$ is the ith element of $T$ when listed in increasing order. For example, if $T = \{1, 3, 5\}$ then the indexed alternating sum of $T$ is \[ 1 \cdot 1 - 2 \cdot 3 + 3 \cdot 5 = 10. \] Alternating sums of empty sets are defined to be $0$.

There are $n$ blue points and $n$ red points on a straight line. Prove that the sum of all distances between pairs of points of the same colour is less than or equal to the sum of all distances between pairs of points of different colours

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ such that $ a\plus{}b\equal{}c\plus{}d\plus{}e$: \[ \sqrt{a^{2}\plus{}b^{2}\plus{}c^{2}\plus{}d^{2}\plus{}e^{2}}\geq T(\sqrt a\plus{}\sqrt b\plus{}\sqrt c\plus{}\sqrt d\plus{}\sqrt e)^{2}\]

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows: [list] [*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and [*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$. [/list] Show that $a_n$ is always a strictly positive integer.

Find all real polynomials $ f(x)$ satisfying $ f(x^2)\equal{}f(x)f(x\minus{}1)$ for all $ x$.

Compute the number of ways to divide a $20 \times 24 $ rectangle into $4 \times 5$ rectangles. (Rotations and reflections are considered distinct.)

Let $a$ be number of $n$ digits ($ n > 1$). A number $b$ of $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values values of $k$.

Let $n$ be a positive integer. On a blackboard is a circle, and around it $n$ non-negative integers are written. Raphinha plays a game in which an operation consists of erasing a number $a$ neighboring $b$ and $c$, with $b \geq c$, and writing in its place $b + c$ if $b + c \leq 5a/4$ and $b - c$ otherwise. Your goal is to make all the numbers on the board equal $0$. Find all $n$ such that Raphinha always manages to reach her goal.

For a given positive integer $n(\ge 2)$, find maximum positive integer $A$ such that there exists $P \in \mathbb{Z}[x]$ with degree $n$ that satisfies the following two conditions. [list] [*] For any $1 \le k \le A$, it satisfies that $A \mid P(k)$, and [*] $P(0)= 0$ and the coefficient of the first term of $P$ is $1$, which means that $P(x)$ is in the following form where $c_2, c_3, \cdots, c_n$ are all integers and $c_n \neq 0$. $$P(x) = c_nx^n + c_{n-1}x^{n-1}+\dots+c_2x^2+x$$ [/list]

$a_1=-1$, $a_2=2$, and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$. What is $a_{2006}$? $ \textbf{(A)}\ -2 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ -\frac 12 \qquad\textbf{(D)}\ \frac 12 \qquad\textbf{(E)}\ 2 $

In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$? \[\begin{array}{lr} &ABBCB \\ +& BCADA \\ \hline & DBDDD \end{array}\] $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Let \( n \geq 4 \) be a positive integer. Initially, each of \( n \) girls knows one piece of gossip that no one else knows, and they want to share them. For greater security, to avoid being spied, they only talk in pairs, and in a conversation, each girl shares all the gossip she knows so far with the other one. What is the minimum number of conversations needed so that every girl knows all the gossip?

An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

Determine the value of $-1+2+3+4-5-6-7-8-9+...+10000$, where the signs change after each perfect square. [i]Proposed by Michael Ren

Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?

Let K and D be the set of convergent and divergent series of positive terms respectively. Does there exist a bijection between K and D such that for all $\sum a_n,\sum b_n\in K$ and $\sum a_n',\sum b_n'\in D$ , $\frac{a_n}{b_n}\to 0\iff \frac{a_n'}{b_n'}\to 0$ ? Under the bijection, $\sum a_n\leftrightarrow\sum a_n'$ and $\sum b_n\leftrightarrow\sum b_n'$.

Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$. ([i]Dan Schwarz[/i])

In a triangle $ABC$, the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$, respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$. The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$, and the length of their common chord is denoted by $m$. Prove that $PQ = m$