A beam of light strikes $\overline{BC}$ at point $C$ with angle of incidence $\alpha=19.94^\circ$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection. Given that $\beta=\alpha/10=1.994^\circ$ and $AB=AC,$ determine the number of times the light beam will bounce off the two line segments. Include the first reflection at $C$ in your count. [asy] size(250);defaultpen(linewidth(0.7)); real alpha=24, beta=32; pair B=origin, C=(1,0), A=dir(beta), D=C+0.5*dir(alpha); pair EE=2*dir(180-alpha), E=intersectionpoint(C--EE, A--B); pair EEE=reflect(B,A)*EE, EEEE=reflect(C,B)*EEE, F=intersectionpoint(E--EEE, B--C), G=intersectionpoint(F--EEEE, A--B); draw((1.4,0)--B--1.4*dir(beta)); draw(D--C, linetype("4 4"),EndArrow(5)); draw(C--E, linetype("4 4"),EndArrow(5)); draw(E--F, linetype("4 4"),EndArrow(5)); draw(F--G, linetype("4 4"),EndArrow(5)); markscalefactor=0.01; draw(anglemark(C,B,A)); draw(anglemark((1.4,0), C,D)); label("$\beta$", 0.07*dir(beta/2), dir(beta/2), fontsize(10)); label("$\alpha$", C+0.07*dir(alpha/2), dir(alpha/2), fontsize(10)); label("$A$", A, dir(90)*dir(A)); label("$B$", B, dir(beta/2+180)); label("$C$", C, S);[/asy]

Evaluate \[ \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.\]

(M.Volchkevich, 9--11) Given two circles and point $ P$ not lying on them. Draw a line through $ P$ which cuts chords of equal length from these circles.

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent. [i]Proposed by Eugene Bilopitov, Ukraine[/i]

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)

Let $ ABCDE$ be convex pentagon such that $ S(ABC) \equal{} S(BCD) \equal{} S(CDE) \equal{} S(DEA) \equal{} S(EAB)$. Prove that there is a point $ M$ inside pentagon such that $ S(MAB) \equal{} S(MBC) \equal{} S(MCD) \equal{} S(MDE) \equal{} S(MEA)$.

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.) b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?

Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$, $ AB\equal{}11$, $ BC\equal{}5$, $ CD\equal{}19$, and $ DA\equal{}7$. Bisectors of $ \angle A$ and $ \angle D$ meet at $ P$, and bisectors of $ \angle B$ and $ \angle C$ meet at $ Q$. What is the area of hexagon $ ABQCDP$? $ \textbf{(A)}\ 28\sqrt{3}\qquad \textbf{(B)}\ 30\sqrt{3}\qquad \textbf{(C)}\ 32\sqrt{3}\qquad \textbf{(D)}\ 35\sqrt{3}\qquad \textbf{(E)}\ 36\sqrt{3}$

We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line $ l$ such that the sum of the lengths of the projections of the given segments to the line $ l$ is less than $ \frac{2}{\pi}.$

Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

Find the area of a square inscribed in an equilateral triangle, with one edge of the square on an edge of the triangle, if the side length of the triangle is $ 2\plus{}\sqrt{3}$.

Given triangle $ABC$. Lines $AL_a$ and $AM_a$ are the internal and the external bisectrix of angle $A$. Let $\omega_a$ be the reflection of the circumcircle of $\triangle AL_aM_a$ in the midpoint of $BC$. Circle $\omega_b$ is defined similarly. Prove that $\omega_a$ and $\omega_b$ touch if and only if $\triangle ABC$ is right-angled.

Six cities $A, B, C, D, E$, and $F$ are located on the vertices of a regular hexagon in that order. $G$ is the center of the hexagon. The sides of the hexagon are the roads connecting these cities. Further more, there are roads connecting cities $B, C, E, F$ and $G$, respectively. Because of raining, one or more roads maybe destroyed. The probability of the road keeping undestroyed between two consecutive cities is $p$. Determine the probability of the road between cities $A$ and $D$ is undestroyed.

Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$

Let $ABCD$ be a regular tetrahedron with side length $2$. The plane parallel to edges $AB$ and $CD$ and lying halfway between them cuts $ABCD$ into two pieces. Find the surface area of one of these pieces.

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

Let a positive integer $n$.Consider square table $3*3$.One use $n$ colors to color all cell of table such that each cell is colored by exactly one color. Two colored table is same if we can receive them from other by a rotation through center of $3*3$ table How many way to color this square table satifies above conditions.

Determine the value of $ab$ if $\log_8 a + \log_4 b^2 = 5$ and $\log_8 b + \log_4 a^2 = 7$.

Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy? [i]Proposed by A. Golovanov[/i]

Let $A_1A_2A_3\dots A_{20}$ be a $20$-sided polygon $P$ in the plane, where all of the side lengths of $P$ are equal, the interior angle at $A_i$ measures $108$ degrees for all odd $i$, and the interior angle $A_i$ measures $216$ degrees for all even $i$. Prove that the lines $A_2A_8$, $A_4A_{10}$, $A_5A_{13}$, $A_6A_{16}$, and $A_7A_{19}$ all intersect at the same point. [asy] import graph; size(10cm); pair temp= (-1,0); pair A01 = (0,0); pair A02 = rotate(306,A01)*temp; pair A03 = rotate(144,A02)*A01; pair A04 = rotate(252,A03)*A02; pair A05 = rotate(144,A04)*A03; pair A06 = rotate(252,A05)*A04; pair A07 = rotate(144,A06)*A05; pair A08 = rotate(252,A07)*A06; pair A09 = rotate(144,A08)*A07; pair A10 = rotate(252,A09)*A08; pair A11 = rotate(144,A10)*A09; pair A12 = rotate(252,A11)*A10; pair A13 = rotate(144,A12)*A11; pair A14 = rotate(252,A13)*A12; pair A15 = rotate(144,A14)*A13; pair A16 = rotate(252,A15)*A14; pair A17 = rotate(144,A16)*A15; pair A18 = rotate(252,A17)*A16; pair A19 = rotate(144,A18)*A17; pair A20 = rotate(252,A19)*A18; dot(A01); dot(A02); dot(A03); dot(A04); dot(A05); dot(A06); dot(A07); dot(A08); dot(A09); dot(A10); dot(A11); dot(A12); dot(A13); dot(A14); dot(A15); dot(A16); dot(A17); dot(A18); dot(A19); dot(A20); draw(A01--A02--A03--A04--A05--A06--A07--A08--A09--A10--A11--A12--A13--A14--A15--A16--A17--A18--A19--A20--cycle); label("$A_{1}$",A01,E); label("$A_{2}$",A02,W); label("$A_{3}$",A03,NE); label("$A_{4}$",A04,SW); label("$A_{5}$",A05,N); label("$A_{6}$",A06,S); label("$A_{7}$",A07,N); label("$A_{8}$",A08,SE); label("$A_{9}$",A09,NW); label("$A_{10}$",A10,E); label("$A_{11}$",A11,W); label("$A_{12}$",A12,E); label("$A_{13}$",A13,SW); label("$A_{14}$",A14,NE); label("$A_{15}$",A15,S); label("$A_{16}$",A16,N); label("$A_{17}$",A17,S); label("$A_{18}$",A18,NW); label("$A_{19}$",A19,SE); label("$A_{20}$",A20,W);[/asy]