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.)

Let $ n$ be a given positive integer. In the coordinate set, consider the set of points $ \{P_{1},P_{2},...,P_{4n\plus{}1}\}\equal{}\{(x,y)|x,y\in \mathbb{Z}, xy\equal{}0, |x|\le n, |y|\le n\}.$ Determine the minimum of $ (P_{1}P_{2})^{2} \plus{} (P_{2}P_{3})^{2} \plus{}...\plus{} (P_{4n}P_{4n\plus{}1})^{2} \plus{} (P_{4n\plus{}1}P_{1})^{2}.$

Let $ABCD$ be a trapezoid with $AB \parallel CD, AB=20, CD=24,$ and area $880.$ Compute the area of the triangle formed by the midpoints of $AB, AC,$ and $BD.$

A point $X$ is the origin of three rays such that the angle between any two of them equals $120^{\circ}$. Let $\omega$ be an arbitrary circle with radius $R$ such that $X$ lies inside it, and $A$, $B$, $C$ be the common points of the rays with this circle. Find $max(XA+XB+XC)$. Proposed by: F.Nilov

A circle is tangent to sides $AD,\text{ }DC,\text{ }CB$ of a convex quadrilateral $ABCD$ at $\text{K},\text{ L},\text{ M}$ respectively. A line $l$, passing through $L$ and parallel to $AD$, meets $KM$ at $N$ and $KC$ at $P$. Prove that $PL=PN$.

Can the equation $x^3 + ax^2 + bx + c = 0$ have only negative roots , if we know that $a+2b+4c=- \frac12 $?

Find the sum of all real $x$ such that \[\frac{4x^2 + 15x + 17}{x^2 + 4x + 12}=\frac{5x^2 + 16x + 18}{2x^2 + 5x + 13}.\]

Laura has $2010$ lamps connected with $2010$ buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons simultaneously, so the lamps are lit simultaneously, too. a) If Richard chooses the buttons to be pressed, what is the maximum number of different combinations of buttons he can press until Laura can assign the buttons to the lamps correctly? b) Supposing that Laura will choose the combinations of buttons to be pressed, what is the minimum number of attempts she has to do until she is able to associate the buttons with the lamps in a correct way?

Circles $ \omega_1$ and $ \omega_2$ meet at $ P$ and $ Q$. Segments $ AC$ and $ BD$ are chords of $ \omega_1$ and $ \omega_2$ respectively, such that segment $ AB$ and ray $ CD$ meet at $ P$. Ray $ BD$ and segment $ AC$ meet at $ X$. Point $ Y$ lies on $ \omega_1$ such that $ PY \parallel BD$. Point $ Z$ lies on $ \omega_2$ such that $ PZ \parallel AC$. Prove that points $ Q,X,Y,Z$ are collinear.

What is the measure of the largest convex angle formed by the hour and minute hands of a clock between $1:45$ PM and $2:40$ PM, in degrees? Convex angles always have a measure of less than $180$ degrees.

Two equal polygons $F$ and $F'$ are given on the plane. It is known that the vertices of the polygon $F$ belong to $F'$ (may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?

In the given sequence $1,4,8,10,16,19,21,25,30,43$, sum of a few adjacent numbers in the sequence is a multiple of $11$. The number of such number sets is________.

Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$. (1) Find $ I_0,\ I_{\minus{}1},\ I_2$. (2) Find $ I_1$. (3) Express $ I_{n\plus{}2}$ in terms of $ I_n$. (4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$. (5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results. You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.

Let $w$ be a circle and $AB$ a line not intersecting $w$. Given a point $P_{0}$ on $w$, define the sequence $P_{0},P_{1},\ldots $ as follows: $P_{n\plus{}1}$ is the second intersection with $w$ of the line passing through $B$ and the second intersection of the line $AP_{n}$ with $w$. Prove that for a positive integer $k$, if $P_{0}\equal{}P_{k}$ for some choice of $P_{0}$, then $P_{0}\equal{}P_{k}$ for any choice of $P_{0}$. [i]Gheorge Eckstein[/i]

Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let \[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\] Prove that : $a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b]. $b)$ the number of good subsets of $T$ is [b]odd[/b].

Determine all positive integers $ n$ for which there is a coloring of all points in space so that each of the following conditions is satisfied: (i) Each point is painted in exactly one color. (ii) Exactly $ n$ colors are used. (iii) Each line is painted in at most two different colors.

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

In triangle $ABC$, the angle bisectors are $BE$ and $CF$ (where $E, F$ are on the sides of the triangle), and their intersection point is $I$. Point $N$ lies on the circumcircle of $AEF$, and the angle $\angle IAN$ is right. The circumcircle of $AEF$ meets the line $NI$ a second time at the point $L$. Show that the circumcenter of $AIL$ lies on line $BC$.

Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$? $\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$

Suppose $S$ is the set of all positive integers. For $a,b \in S$, define \[a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}\] For example $8*12=6$. Show that [b]exactly two[/b] of the following three properties are satisfied: (i) If $a,b \in S$, then $a*b \in S$. (ii) $(a*b)*c=a*(b*c)$ for all $a,b,c \in S$. (iii) There exists an element $i \in S$ such that $a *i =a$ for all $a \in S$.

Let $a,b,c,d$ be the four roots of the polynomial \[x^4+3x^3-x^2+x-2.\] Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{2}$ and $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}=-\frac{3}{4}$, the value of \[\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Let $\omega$ be the circumcircle of triangle $ABC$. A point $B_1$ is chosen on the prolongation of side $AB$ beyond point B so that $AB_1 = AC$. The angle bisector of $\angle BAC$ meets $\omega$ again at point $W$. Prove that the orthocenter of triangle $AWB_1$ lies on $\omega$ . (A.Tumanyan)