Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circumcircle of $\triangle{BOC}$ meets the lines $AB, AC$ at points $A_1, A_2$, respectively. Let $\omega_{A}$ be the circumcircle of triangle $AA_1A_2$. Define $\omega_B$ and $\omega_C$ analogously. Prove that the circles $\omega_A, \omega_B, \omega_C$ concur on $\odot(ABC)$.

Nine tiles are numbered $1, 2, 3, \ldots, 9,$ respectively. Each of three players randomly selects and keeps three of the tile, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

An equilateral triangle of altitude $1$ is considered. For every point $P$ on the interior of the triangle, denote by $x, y , z$ the distances from the point $P$ to the sides of the triangle. a) Prove that for every point $P$ inside the triangle it is true that $x + y + z = 1$. b) For which points of the triangle does it hold that the distance to one side is greater than the sum of the distances to the other two? c) We have a bar of length $1$ and we break it into three pieces. find the probability that with these pieces a triangle can be formed.

$ABC$ is a triangle and $D$ is an internal point such that $\angle DAB=\angle DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.

Given a real number $\alpha$ in whose decimal representation $\alpha=0,a_1a_2a_3\dots$ each decimal digit $a_i$ $(i=1,2,3,\dots)$ is a prime number. The decimal digits are arranged along the path indicated by arrows in the accompanying figure, which can be thought of as continuing infinitely to the right and downward. For each $m\geq 1$, the decimal representation of a real number $z_m$ is formed by writing before the decimal point the digit 0 and after the decimal point the sequence of digits of the $m$-th row from the top read from left to right from the adjacent arrangement. In an analogous way, for all $n\geq 1$, the real numbers $s_n$ are formed with the digits of the $n$-th column from the left to be read from top to bottom. For example, $z_3=0,a_5a_6a_7a_{12}a_{23}a_{28}\dots$ and $s_2=0,a_2a_3a_6a_{15}a_{18}a_{35}\dots$. Show: (a) If $\alpha$ is rational, then all $z_m$ and all $s_n$ are rational. (b) The converse of the statement formulated in (a) is false.

For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$

Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

Find all pair $(x,y)$ of positive integers satisfying $$\left|\frac{x}{y}-\sqrt2\right|<\frac{1}{y^3}.$$

Given is an obtuse isosceles triangle $ABC$ with $CA=CB$ and circumcenter $O$. The point $P$ on $AB$ is such that $AP<\frac{AB} {2}$ and $Q$ on $AB$ is such that $BQ=AP$. The circle with diameter $CQ$ meets $(ABC)$ at $E$ and the lines $CE, AB$ meet at $F$. If $N$ is the midpoint of $CP$ and $ON, AB$ meet at $D$, show that $ODCF$ is cyclic.

Prove that $$\sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} &lt; 2.$$ Proposed by Ivan Krijan, University of Zagreb

Consider triangle $ABC$ with sides $AB=4$, $BC=11$, and $CA=9$. The triangle is spun around a line that passes through $B$ and the interior of the triangle (including the edges $BC$ and $BA$). Of all possible lines with these constraints, what is the largest possible volume of the resulting solid?

Given four points $ A_1,A_2,A_3,A_4$ in the plane in such a way that $ A_4$ is the centroid of the $ \bigtriangleup A_1A_2A_3$, find a point $ A_5$ in the plane that maximizes the ratio \[ \frac{\min_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}{\max_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}.\] ($ T(ABC)$ denotes the area of the triangle $ \bigtriangleup ABC.$ ) [i]J. Suranyi[/i]

Given a convex quadrilateral $ABCD$. We denote $I_A,I_B, I_C$ and $I_D$ centers of $\omega_A, \omega_B,\omega_C $and $\omega_D$,inscribed In the triangles $DAB, ABC, BCD$ and $CDA$, respectively.It turned out that $\angle BI_AA + \angle I_CI_AI_D = 180^\circ$. Prove that $\angle BI_BA + \angle I_CI_BI_D = 180^{\circ}$. (A. Kuznetsov)

Is it possible to paint $33$ squares on a $9\times 9$ game board, so that each row and each column of the board has a maximum of $4$ painted squares, but if we also paint any other square a row or column appears that has $5$ squares painted?

Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively. 30 points

Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.) Does the same result hold with a degree 3 or degree 5 polynomial?

Which pair of aqueous solutions produce a yellow precipitate upon mixing? $ \textbf{(A)} \hspace{.05in} \ce{AlCl3} \text{ and } \ce{KOH} \qquad \textbf{(B)}\hspace{.05in}\ce{Ba(NO3)2} \text{ and }\ce{ Na2SO4} \qquad $ $\textbf{(C)}\hspace{.05in}\ce{Cu(NO3)2} \text{ and } \ce{NaClO4} \qquad \textbf{(D)}\hspace{.05in}\ce{Pb(C2H3O2)2} \text{ and } \ce{KI} \qquad$

If $ x$ is a real number such that $ x^2\minus{}x$ and $ x^n\minus{}x$ are integers for some $ n \ge 3$, prove that $ x$ is an integer.

Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, $i = 1, 2, \dots, m$. It is known that every two elements of $X$ is contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.

Let $ABC$ be a triangle with $AB=AC$. A circle $\Gamma$ lies outside triangle $ABC$ and is tangent to line $AC$ at $C$. Point $D$ lies on $\Gamma$ such that the circumcircle of triangle $ABD$ is internally tangent to $\Gamma$. Segment $AD$ meets $\Gamma$ secondly at $E$. Prove that $BE$ is tangent to $\Gamma$

Consider a triangle $\Delta ABC$ and three points $D,E,F$ such that: $B$ and $E$ are on different side of the line $AC$, $C$ and $D$ are on different sides of $AB$, $A$ and $F$ are on the same side of the line $BC$. Also $\Delta ADB \sim \Delta CEA \sim \Delta CFB$. Let $M$ be the middle point of $AF$. Prove that: a)$\Delta BDF \sim \Delta FEC$. b) $M$ is the middle point of $DE$. [i]Dan Branzei[/i]

Jerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis. Dennis then announces, "Aaron's number is at least three times Jerry's number." Aaron says, "I still don't know Jerry's number." Jerry then replies, "Oh, now I know Aaron's number." What is the sum of their numbers? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

Show that for every natural number $ a\ge 3, $ there are infinitely many natural numbers $ n $ such that $ a^n\equiv 1\pmod n . $ Does this hold for $ n=2? $

Prove that if m,n are nonnegative integers and 0<=x<=1 then $(1-x^n)^m + (1-(1-x)^m)^n \ge 1$

There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac23 \qquad \textbf{(E)}\ \frac34$