Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\omega_2$. Let $P$ be an arbitrary point lying on $\omega_1$. Lines $BP, AP$ and $O_1O_2$ cut $\omega_2$ for the second time at points $X$, $Y$ and $C$, respectively. Prove that quadrilateral $XPYC$ is a parallelogram. [i]Proposed by Iman Maghsoudi[/i]

Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.

A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.

Consider triangle $\vartriangle ABC$ with circumradius $R = 10$, inradius $r = 2$ and semi-perimeter $S = 18$. Let $I$ be the incenter, and we extend $AI$, $BI$ and $CI$ to intersect the circumcircle at $D, E$ and $F$ respectively. Compute the area of $\vartriangle DEF$.

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ miles per hour, and Joe can run at a speed of $10$ miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance? $\textbf{(A) }7.2\hspace{14em}\textbf{(B) }14.4\hspace{14em}\textbf{(C) }36$ $\textbf{(D) }10\hspace{14.3em}\textbf{(E) }12\hspace{14.8em}\textbf{(F) }2.4$ $\textbf{(G) }25.2\hspace{13.5em}\textbf{(H) }123456789$

Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\) \[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1p}\\ a_{21} & a_{22} & \ldots & a_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{array}\] such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\). Find \(N \pmod{p}\).

Let $a_1$, $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$. Call a nonempty contiguous subsequence of this sequence [i]good[/i] if the product of the integers in it leaves a remainder of $1$ when divided by $101$. In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible $(a_i)$. [i]Proposed by Yang Liu[/i]

Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $ab$ is not.

Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BP D$?

There are $n\ge 3$ students in a classroom. Every day, the teacher separates the students into at least two non-empty groups, and each pair of students from the same group will shake hands once. Suppose after $k$ days, each pair of students has shaken hands exactly once, and $k$ is as minimal as possible. Prove that $$\sqrt{n} \le k-1 \le 2\sqrt{n}$$ [i]Proposed by Wong Jer Ren[/i]

Let $\angle ABC=24^\circ$ and $\angle ABD =20^\circ$. What is the smallest possible degree measure for $\angle CBD$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 12 $

\[\int_0^{\frac \pi 2} \left(\frac{1}{1-\cos(x)}-\frac{2}{x^2}\right)\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

There are $ 90$ contestants in a mathematics competition. Each contestant gets acquainted with at least $ 60$ other contestants. One of the contestants, Amin, state that at least four contestants have the same number of new friends. Prove or disprove his statement.

What is the maximum number of coins can be arranged in cells of the table $n \times n$ (each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?

Given is a triangle $ABC$.On the extensions of the side $AB$ we consider points $A_1,B_1$ such that $AB_1=BA_1$ (with $A_1$ lying closer to $B$).On the extensions of the side $BC$ we consider points $B_4,C_4$ such that $CB_4=BC_4$ (with $B_4$ lying closer to $C$).On the extensions of the side $AC$ we consider points $C_1,A_4$ such that $AC_1=CA_4$ (with $C_1$ lying closer to $A$).On the segment $A_1A_4$ we consider points $A_2,A_3$ such that $A_1A_2=A_3A_4=mA_1A_4$ where $0<m<\frac{1}{2}$.Points $B_2,B_3$ and $C_2,C_3$ are defined similarly,on the segments $B_1B_4,C_1C_4$ respectively.If $D\equiv BB_2\cap CC_2 \ , \ E\equiv AA_3\cap CC_2 \ , \ F\equiv AA_3\cap BB_3$, $\ G\equiv BB_3\cap CC_3 \ , \ H\equiv AA_2\cap CC_3$ and $I\equiv AA_2\cap BB_2$,prove that the diagonals $DG,EH,FI$ of the hexagon $DEFGHI$ are concurrent. 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label("$C_1$", (0.1036318128096586,5.714897604245849), NE * labelscalefactor); dot((8.151100081632526,-0.9140142233976905),linewidth(3.pt) + dotstyle); label("$A_4$", (8.521999124602214,-1.1903644717669786), NE * labelscalefactor); dot((-2.308476341169285,1.3881159854868106),linewidth(3.pt) + dotstyle); label("$C_3$", (-2.9776006673235647,1.7808239912186203), NE * labelscalefactor); dot((-0.7646359770779035,4.164347956460432),linewidth(3.pt) + dotstyle); label("$C_2$", (-1.1618743843879151,4.504413415622086), NE * labelscalefactor); dot((1.6890544250513695,-1.624864820496926),linewidth(3.pt) + dotstyle); label("$A_2$", (1.6167370485893664,-2.125738617521704), NE * labelscalefactor); dot((5.997084862772141,-1.150964422430769),linewidth(3.pt) + dotstyle); label("$A_3$", (6.211074764502297,-1.603029536070534), NE * labelscalefactor); dot((7.966133662513563,1.6250661845198895),linewidth(3.pt) + dotstyle); label("$B_3$", (8.081823056011753,1.7808239912186203), NE * labelscalefactor); dot((3.7376079411107392,4.8751985535596685),linewidth(3.pt) + dotstyle); 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The bisector of the angle $BAC$of the acute triangle $ABC$ ( $AC \ne AB$) intersects its circumscribed circle for the second time at the point $W$. Let $O$ be the center of the circumscribed circle $\Delta ABC$. The line $AW$ intersects for the second time the circumcribed circles of triangles $OWB$ and $OWC$ at the points $N$ and $M$, respectively. Prove that $BN + MC = AW$. (Mitrofanov V., Hilko D.)

Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$, respectively; let $AD$ be an altitude in this triangle. A point $K$ is chosen on the segment $MN$ so that $BK=CK$. The ray $KD$ meets the circumcircle $\Omega$ of $ABC$ at $Q$. Prove that $C, N, K, Q$ are concyclic.

Suppose that $(a_1,\ldots,a_{20})$ and $(b_1,\ldots,b_{20})$ are two sequences of integers such that the sequence $(a_1,\ldots,a_{20},b_1,\ldots,b_{20})$ contains each of the numbers $1,\ldots,40$ exactly once. What is the maximum possible value of the sum \[\sum_{i=1}^{20}\sum_{j=1}^{20}\min(a_i,b_j)?\]

Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$.

In the bottom-left corner of a chessboard (with 8 rows and 8 columns), there is a king. Marius and Alexandru play a game, with Alexandru going first. On their turn, each player moves the king either one square to the right, one square up, or one square diagonally up-right. The player who moves the king to the top-right corner square wins. Who will win if both players play optimally?

Tom can run to Beth's house in $63$ minutes. Beth can run to Tom's house in $84$ minutes. At noon Tom starts running from his house toward Beth's house while at the same time Beth starts running from her house toward Tom's house. When they meet, they both run at Beth's speed back to Beth's house. At how many minutes after noon will they arrive at Beth's house?

For how many positive integers $n$, $1\leq n\leq 2008$, can the set \[\{1,2,3,\ldots,4n\}\] be divided into $n$ disjoint $4$-element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset?

An integer lattice point in the Cartesian plane is a point $(x,y)$ where $x$ and $y$ are both integers. Suppose nine integer lattice points are chosen such that no three of them lie on the same line. Out of all 36 possible line segments between pairs of those nine points, some line segments may contain integer lattice points besides the original nine points. What is the minimum number of line segments that must contain an integer lattice point besides the original nine points? Prove your answer.

Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior. What is the maximal possible number of points with this property?