[asy] size(200); pair A=(-2,0),B,C=(-1,0),D=(1,0),EE,O=(0,0); draw(arc(O,1, 0, 180)); EE=midpoint(arc(O,1, 0, 90)); draw(A--EE); draw(A--D); B=intersectionpoint(arc(O,1, 180, 0),EE--A); draw(O--EE); label("$A$",A,W); label("$B$",B,NW); label("$C$",C,S);label("$D$",D,E);label("$E$",EE,NE);label("$O$",O,S);label("$45^\circ$",(0.25,0.1),fontsize(10pt)); //Credit to TheMaskedMagician for the diagram [/asy] In the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $AB$ equals length $OD$, and the measure of $\measuredangle EOD$ is $45^\circ$, then the measure of $\measuredangle BAO$ is $\textbf{(A) }10^\circ\qquad\textbf{(B) }15^\circ\qquad\textbf{(C) }20^\circ\qquad\textbf{(D) }25^\circ\qquad\textbf{(E) }30^\circ$

A 3-dimensional cross is made up of 7 cubes, one central cube and 6 cubes that share a face with it. The cross is inscribed in a circle with radius 1. What's its volume?

The cells of the $9 \times 9$ table are colored black and white. It turned out, that there were $k$ rows, in each of which there are more black cells than white ones white, and there were $k$ columns, each of which contained more than black. At what highest $ k$ is this possible?

Let $I$ be the incenter and $AB$ the shortest side of the triangle $ABC$. The circle centered at $I$ passing through $C$ intersects the ray $AB$ in $P$ and the ray $BA$ in $Q$. Let $D$ be the point of tangency of the $A$-excircle of the triangle $ABC$ with the side $BC$. Let $E$ be the reflection of $C$ with respect to the point $D$. Prove that $PE\perp CQ$.

Let $S_0=0$ and let $S_k$ equal $a_1+2a_2+\ldots+ka_k$ for $k\geq 1$. Define $a_i$ to be $1$ if $S_{i-1}<i$ and $-1$ if $S_{i-1}\geq i$. What is the largest $k\leq 2010$ such that $S_k=0$?

A convex $n$-vertex polygon is partitioned into triangles by nonintersecting diagonals . The following operation, called perestroyka (=reconstruction) , is allowed: two triangles $ABD$ and $BCD$ with a common side may be replaced by the triangles $ABC$ and $ACD$. By $P(n)$ denote the smallest number of perestroykas needed to transform any partitioning into any other one. Prove that (a) $P ( n ) \ge n - 3$ (b) $P (n) \le 2n - 7$ (c) $P(n) \le 2n - 10$ if $n \ge 13$ . ( D.Fomin , based on ideas of W. Thurston , D . Sleator, R. Tarjan)

Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.

$ABCDE$ is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments $AC$ and $BD$?

The audience chooses two of twenty-nine cards, numbered from $1$ to $29$ respectively. The assistant of a magician chooses two of the remaining twenty-seven cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in an arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.

In an acute triangle $ABC$, $\angle C = 60^o$. If $AA'$ and $BB'$ are two of the altitudes and $C_1$ is the midpoint of $AB$, prove that triangle $C_1A'B'$ is equilateral.

Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.

Solve the equation $$\frac{\pi-2}{2} + \frac{2}{1+\sin (2\sqrt{x})}+arccos(x^3-8x-1)=tg^2\sqrt{x}- \sqrt{x^4+x^3-5x^2-8x-24}$$

Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?

Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$ [i]Proposed by Christopher Bradley, United Kingdom[/i]

Postive real numbers $a, b, c$ satisfy $abc=1$. Show that $$\frac{a^3+a^2}{1+bc}+\frac{b^3+b^2}{1+ca}+\frac{c^3+c^2}{1+ab}\geq3.$$

Let $a_{1},a_{2},\ldots$ be an infinite arithmetic progression. It's known that there exist positive integers $p,q,t$ such that $a_{p}+tp=a_{q}+tq$. If $a_{t}=t$ and the sum of the first $t$ numbers in the sequence is $18$, determine $a_{2008}$.

If we fill $1\sim 16$ into $4\times4$ chessboard randomly. What is the possibility of the sum of each rows and columns are all even?

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

Consider positive integers $2,3,\ldots,n-1,n$ where $n\ge96.$ Consider any partition in two (sub)sets. Show that at least one of these two sets always contains two numbers and their product. Show that the statement does not hold for $n=95,$ e.g. there is a partition without the mentioned property.

Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$.

Let $T$ be a point on a line segment $AB$ such that $T$ is closer to $B$ than to $A$. Show that for each point $C \ne T$ on the line through $T$ perpendicular to $AB$ there is exactly one point $D$ on the line segment $AC$ with $\angle CBD=\angle BAC$. Moreover, show that the line through $D$ perpendicular to $AC$ intersects the line $AB$ in a point $E$ which is independent of the position of $C$.

Problems 15, 16, and 17 all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles. 17. In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support? $ \text{(A)}\ 50\text{ yrs.}\qquad\text{(B)}\ 75\text{ yrs.}\qquad\text{(C)}\ 100\text{ yrs.}\qquad\text{(D)}\ 125\text{ yrs.}\qquad\text{(E)}\ 150\text{ yrs.} $

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and $$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$ Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points. (Note: a scalene triangle is one where no two sides have equal length.) [i]Proposed by Ankan Bhattacharya, USA[/i]

A $4$ x $4$ square board is called $brasuca$ if it follows all the conditions: • each box contains one of the numbers $0, 1, 2, 3, 4$ or $5$; • the sum of the numbers in each line is $5$; • the sum of the numbers in each column is $5$; • the sum of the numbers on each diagonal of four squares is $5$; • the number written in the upper left box of the board is less than or equal to the other numbers the board; • when dividing the board into four $2$ × $2$ squares, in each of them the sum of the four numbers is $5$. How many $"brasucas"$ boards are there?