Let $ABC$ be an acute triangle with $AB<AC<BC$, inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Let $O_1$ be the symmetric point of $O$ wrt $AC$. Circle $c_1(O_1,R)$ intersects $BC$ at $Z$. If the extension of the altitude $AD$ intersects the cicrumscribed circle $c(O,R)$ at point $E$, prove that $EC$ is perpendicular on $AZ$.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$. let $CE$ be the perpendicular from $C$ on $AB$ prove that $ CE^2 = AB. CD $

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.

If $A$ and $B$ are points of the plane, then by $A * B$ we denote a point symmetric to $A$ with respect to $B$. Is it possible, by applying the operation $*$ several times, to obtain from the three vertices of a given square its fourth vertex?

$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?

A word of length $n$ is an ordered sequence $x_1x_2\ldots x_n$ where $x_i$ is a letter from the set $\{ a,b,c \}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2,$ contains all the letters $a,b,c$. Prove that $|B_{n+1}|=3|A_n|$. [i]Vasile Pop[/i]

Compute the minimum possible value of $(x-1)^2+(x-2)^2+(x-3)^2+(x-4)^2+(x-5)^2$ For real values $x$

Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$. [b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$. [b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.

In a triangle $ABC$, the points $A_1, B_1, C_1$ are symmetric to $A, B,C$ with respect to $B,C, A$, respectively. Given the points $A_1, B_1,C_1$ reconstruct the triangle $ABC$.

Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$. [i]Dinu Șerbănescu[/i]

Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance. The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$ The one who obtains the progression wins. Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$

Let $ABC$ be a triangle with orthocenter $H$. Let $P,Q,R$ be the reflections of $H$ with respect to sides $BC,AC,AB$, respectively. Show that $H$ is incenter of $PQR$.

Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon. [asy] size(8cm); defaultpen(fontsize(10pt)); pair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705); filldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8)); draw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle); pair O = (A_1+A_2+A_3+A_4+A_5)/5; label("$A_1$",A_1, 2dir(A_1-O)); label("$A_2$",A_2, 2dir(A_2-O)); label("$A_3$",A_3, 2dir(A_3-O)); label("$A_4$",A_4, 2dir(A_4-O)); label("$A_5$",A_5, 2dir(A_5-O)); label("$B_1$",B_1, 2dir(B_1-O)); label("$B_2$",B_2, 2dir(B_2-O)); label("$B_3$",B_3, 2dir(B_3-O)); label("$B_4$",B_4, 2dir(B_4-O)); label("$B_5$",B_5, 2dir(B_5-O)); [/asy]

An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A [i]tour route[/i] is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. [asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy] How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? [list] [*] some rotation around a point of line $\ell$ [*] some translation in the direction parallel to line $\ell$ [*] the reflection across line $\ell$ [*] some reflection across a line perpendicular to line $\ell$ [/list] $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Let $P(A)$ be the arithmetic-means of all elements of set $A = \{ a_1, a_2, \ldots, a_n \}$, namely $P(A) = \frac{1}{n} \sum^{n}_{i=1}a_i$. We denote $B$ "balanced subset" of $A$, if $B$ is a non-empty subset of $A$ and $P(B) = P(A)$. Let set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$. Find the number of all "balanced subset" of $M$.

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$.

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.)

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 $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that (a) $M$ is the midpoint of $AB$; (b) $N$ is the midpoint of $AC$; (c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$. Prove that $\angle APM = \angle PBA$.

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$. [i]Proposed by Allen Liu[/i]

Given is the non-equilateral triangle $A_1A_2A_3$. $B_{ij}$ is the symmetric of $A_i$ wrt the inner bisector of $\angle A_j$. Prove that lines $B_{12}B_{21}$, $B_{13}B_{31}$ and $B_{23}B_{32}$ are parallel.

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