Let $ABCDEFG$ be a regular heptagon. We'll call the sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$ and $GA$ opposite to the vertices $E$, $F$, $G$, $A$, $B$, $C$ and $D$ respectively. If $M$ is a point inside the heptagon, we'll say that the line through $M$ and a vertex of the heptagon intersects a side of it (without the vertices) at a $\textit{perfect}$ point, if this side is opposite the vertex. Prove that for every choice of $M$, the number of $\textit{perfect}$ points is always odd.

In a party of eight people, each pair of people either know each other or do not know each other. Each person knows exactly three of the others. Determine whether the following two conditions can be satisfied simultaneously: [list] – for any three people, at least two do not know each other; – for any four people there are at least two who know each other. [/list]

Prove that there is a unique positive integer formed only by the digits $2$ and $5$, which has $ 2007$ digits and is divisible by $2^{2007}$.

Let $m,n,p$ be rational numbers such that $\sqrt{m} + \sqrt{n} + \sqrt{p}$ is a rational number. Prove that $\sqrt{m}, \sqrt{n}, \sqrt{p}$ are also rational numbers

Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other? $\textbf{(A)}\ 1:24\qquad \textbf{(B)}\ 1:28\qquad \textbf{(C)}\ 1:36\qquad \textbf{(D)}\ 1:40\qquad \textbf{(E)}\ 1:48$

Determine all roots, real or complex, of the system of simultaneous equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}

The number $n{}$ cointains $k{}$ units in binary system. Prove that $2^{n-k}{}$ divides $n!$.

A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)

Let $ABC$ be an equilateral triangle. Let $P$ be a point inside of it such that the square root of the distance of $P$ to one of the sides is equal to the sum of the square roots of the distances of $P$ to the other two sides. Find the geometric place of $P$.

[b]6.[/b] Let $T$ be a one-to-one mapping of the unit square $E$ of the plane into itself. Suppose that $T$ and $T^{-1}$ are measure-preserving (i.e. if $M \subseteq E$ is a measurable set, then $TM$ and $T^{-1}M$ are also measurable and $\mu (M)= \mu (TM)= \mu (T^{-1}M)$, where $\mu$ denotes the Lebesgue measure) and, furthermore, that if $Tx \in N$ for almost all points $x$ of a measurable set $N \subseteq E$, then either $n$ or $ E \setminus N$ is of measure 0. Prove that, for any measurable set $A \subseteq E$, with $\mu (A)>0$, the function $n(x)$ defined by $$n(x)=\begin{cases} 0, \mbox{if} \quad T^k x \notin A \quad (k=1, 2, \dots),\\ \min (k: T^k x \in A; k=1,2, \dots ) &\mbox{otherwise} \end{cases} $$ is measurable and $\int_{A}n(x) d\mu(x) =1$ [b](R. 18)[/b]

Let $a_1=24$ and form the sequence $a_n$, $n\geq 2$ by $a_n=100a_{n-1}+134$. The first few terms are $$24,2534,253534,25353534,\ldots$$ What is the least value of $n$ for which $a_n$ is divisible by $99$?

A set contains $ 372$ integers from $ 1,2,...,1200$ . For every element $ a\in S$, the numbers $ a\plus{}4,a\plus{}5,a\plus{}9$ don't belong to $ S$. Prove that $ 600\in S$.

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

In the following addition, different letters represent different non-zero digits. What is the 5-digit number $ABCDE$? $ \begin{array}{ccccccc} A&B&C&D&E&D&B\\ &B&C&D&E&D&B\\ &&C&D&E&D&B\\ &&&D&E&D&B\\ &&&&E&D&B\\ &&&&&D&B\\ +&&&&&&B\\ \hline A&A&A&A&A&A&A \end{array} $

Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation $$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$ then $S$ consists of one function, the identity function. (a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$. (b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$? *These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$. [i]Proposed by Carl Schildkraut[/i]

Suppose that $m, n, k$ are positive integers satisfying $$3mk=(m+3)^n+1.$$ Prove that $k$ is odd.

A storage depot is a pyramid with height $30$ and a square base with side length $40$. Determine how many cubical $3\times 3\times 3$ boxes can be stored in this depot if the boxes are always packed so that each of their edges is parallel to either an edge of the base or the altitude of the pyramid.

Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$ "chunks" and the channel can transmit $120$ chunks per second. $ \textbf{(A)}\ 0.04 \text{ seconds}\qquad\textbf{(B)}\ 0.4 \text{ seconds}\qquad\textbf{(C)}\ 4 \text{ seconds}\qquad\textbf{(D)}\ 4 \text{ minutes}\qquad\textbf{(E)}\ 4 \text{ hours} $

Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares. a) Give an example of such numbers $a, b$ and $c$. b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$

In how many ways can one fill an $n*n$ matrix with $+1$ and $-1$ so that the product of the entries in each row and each column equals $-1$?

On the blackboard were six figures: a circle, a triangle, a square, a trapezoid, a pentagon and a hexagon, painted in six colors: blue, white, red, yellow, green and brown. Each figure had only one color and all the figures were of different colors. The next day he wondered what color each figure was. Paul replied: “The circle was red, the triangle was blue, the square was white, the trapezoid was green, the pentagon was brown, and the hexagon was yellow. Sofía answered: “The circle was yellow, the triangle was green, the square was red, the trapezoid was blue, the pentagon was brown, and the hexagon was white.” Pablo was wrong three times and Sofia twice, and it is known that the pentagon was brown. Determine if it is possible to know with certainty what the color of each of the figures was.

Prove that among the seven natural numbers forming an arithmetic progression with difference $ 30 $ , one and only one is divisible by $ 7 $ .

Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation \[a^3+2b^3+4c^3=6abc+1.\]

Let $\vartriangle ABC$ be an acute triangle, and let $A'$ and $B'$ be the feet of altitudes from $A$ to $BC$ and from $B$ to $CA$, respectively; the altitudes intersect at $H$. If $BH$ is equal to the circumradius of $\vartriangle ABC$, find $\frac{A'B}{AB}$ .

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.