here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n. if S=s1,s2,s3,.... be a finite or infinite sequence such that for each i, si is in {1,2,...,n}, we move P on the plane according to the S in this form: at first we reflect P through the s1 ( s1 means the edge which iys number is s1)then through s2 and so on like the figure below. a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane b)prove that the sequence in a) isn't periodic. c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]

In an acute-angled triangle, each angle is an integral number of degrees, and the smallest angle is one-fifth of the largest one. Find these angles. (G Galperin)

Let $A, B, C, D$ be points in a line $l$ in this order where $AB = BC$ and $AC = CD$. Let $w$ be a circle that passes in the points $B$ and $D$, a line that passes by $A$ intersects $w$ in the points $P$ and $Q$(the point $Q$ is in the segment $AP$). Let $M$ be the midpoint of $PD$ and $R$ is the symmetric of $Q$ by the line $l$, suppose that the segments $PR$ and $MB$ intersect in the point $N$. Prove that the quadrilateral $PMNC$ is cyclic

Let three circles be externally tangent in pairs, with parallel diameters $A_1A_2, B_1B_2, C_1C_2$ (i.e. each of the quadrilaterals $A_1B_1B_2A_2$ and $A_1C_1C_2A_2$ is a parallelogram or trapezoid, which segment $A_1A_2$ is the base). Prove that $A_1B_2, B_1C_2, C_1A_2$ intersect at one point. (Yuri Biletsky )

Each point of the 3-dimensional space is coloured with exactly one of the colours red, green and blue. Let $R$, $G$ and $B$, respectively, be the sets of the lengths of those segments in space whose both endpoints have the same colour (which means that both are red, both are green and both are blue, respectively). Prove that at least one of these three sets includes all non-negative reals.

Square $A$ is adjacent to square $B$ which is adjacent to square $C$. The three squares all have their bottom sides along a common horizontal line. The upper left vertices of the three squares are collinear. If square $A$ has area $24$, and square $B$ has area $36$, find the area of square $C$. [asy] import graph; size(8cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); real xmin = -4.89, xmax = 13.61, ymin = -1.39, ymax = 9; draw((0,0)--(2,0)--(2,2)--(0,2)--cycle, linewidth(1.2)); draw((2,0)--(5,0)--(5,3)--(2,3)--cycle, linewidth(1.2)); draw((5,4.5)--(5,0)--(9.5,0)--(9.5,4.5)--cycle, linewidth(1.2)); draw((2,0)--(2,2), linewidth(1.2)); draw((2,2)--(0,2), linewidth(1.2)); draw((0,2)--(0,0), linewidth(1.2)); draw((2,0)--(5,0), linewidth(1.2)); draw((5,0)--(5,3), linewidth(1.2)); draw((5,3)--(2,3), linewidth(1.2)); draw((2,3)--(2,0), linewidth(1.2)); draw((5,4.5)--(5,0), linewidth(1.2)); draw((5,0)--(9.5,0), linewidth(1.2)); draw((9.5,0)--(9.5,4.5), linewidth(1.2)); draw((9.5,4.5)--(5,4.5), linewidth(1.2)); label("A",(0.6,1.4),SE*labelscalefactor); label("B",(3.1,1.76),SE*labelscalefactor); label("C",(6.9,2.5),SE*labelscalefactor); draw((13.13,8.56)--(-3.98,0), linewidth(1.2)); draw((-3.98,0)--(15.97,0), linewidth(1.2));[/asy]

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

A circle $\mathcal C$ and five distinct points $M_1,M_2,M_3,M_4$ and $M$ on $\mathcal C$ are given in the plane. Prove that the product of the distances from $M$ to lines $M_1M_2$ and $M_3M_4$ is equal to the product of the distances from $M$ to the lines $M_1M_3$ and $M_2M_4$. What can one deduce for $2n+1$ distinct points $M_1,\ldots,M_{2n},M$ on $\mathcal C$?

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$. Let $X$ be the foot of the altitude from $O$ onto line $PT$. Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$. *The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$, $\overline{BH}$, and $\overline{CH}$. [i]Proposed by Zack Chroman[/i]

A triangle ABC is inscribed in a circle with center $ O$ and radius $ R$. If the inradii of the triangles $ OBC, OCA, OAB$ are $ r_{1}, r_{2}, r_{3}$ , respectively, prove that $ \frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\geq\frac{4\sqrt{3}+6}{R}.$

In $\triangle ABC$, $P$ is a point on $BC$. $F\in AB,E\in AC,PF//CA,PE//BA$. If $S_{\triangle ABC}=1$, prove that at least one of $S_{\triangle BPF},S_{\triangle PCE},S_{PEAF}$ is not less than $\frac{4}{9}$.

Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $

Label the squares of a $1991 \times 1992$ rectangle $(m, n)$ with $1 \leq m \leq 1991$ and $1 \leq n \leq 1992$. We wish to color all the squares red. The first move is to color red the squares $(m, n), (m+1, n+1), (m+2, n+1)$for some $m < 1990, n < 1992$. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?

We are given three points $A,B,C$ on a semicircle. The tangent lines at $A$ and $B$ to the semicircle meet the extension of the diameter at points $M,N$ respectively. The line passing through $A$ that is perpendicular to the diameter meets $NC$ at $R$, and the line passing through $B$ that is perpendicular to the diameter meets $MC$ at $S$. If the line $RS$ meets the extension of the diameter at $Z$, prove that $ZC$ is tangent to the semicircle.

A line $d$ is given in the plane. Let $B\in d$ and $A$ another point, not on $d$, and such that $AB$ is not perpendicular on $d$. Let $\omega$ be a variable circle touching $d$ at $B$ and letting $A$ outside, and $X$ and $Y$ the points on $\omega$ such that $AX$ and $AY$ are tangent to the circle. Prove that the line $XY$ passes through a fixed point. [i]Proposed by F. Bakharev [/i]

In triangle $ABC$ $\omega$ is incircle and $\omega_1$,$\omega_2$,$\omega_3$ is tangents to $\omega$ and two sides of $ABC$. $r, r_1, r_2, r_3$ is radius of $\omega, \omega_1, \omega_2, \omega_3$. Prove that $\sqrt{r_1 r_2}+\sqrt{r_2 r_3}+\sqrt{r_3 r_1}=r$

Quadrilateral $ABCD$ is inscribed in a circle of radius $R$, and also circumscribed around a circle of radius $r$. It is known that $\angle ADB = 45^o$. Find the area of triangle $AIB$, where point $I$ is the center of the circle inscribed in $ABCD$. (Hryhoriy Filippovskyi)

A point $H$ lies on the side $AB$ of regular polygon $ABCDE$. A circle with center $H$ and radius $HE$ meets the segments $DE$ and $CD$ at points $G$ and $F$ respectively. It is known that $DG=AH$. Prove that $CF=AH$.

In a convex quadrilateral $ABCD$, angles $B$ and $D$ are right angles. On on sides $AB$, $BC$, $CD$, $DA$ points $K$, $L$, $M$, $N$ are taken respectively so that $KN \perp AC$ and $LM \perp AC$. Prove that $KM$, $LN$ and $AC$ intersect at one point.

Given a triangle $ABC$ where $|BC| = a, |CA| = b$ and $|AB| = c$, prove that the equality $\frac{1}{a + b}+\frac{1}{b + c}=\frac{3}{a + b + c}$ holds if and only if $\angle ABC = 60^o$.

The numbers $S_1=2^2, S_2=2^4,\ldots, S_n=2^{2n}$ are given. A rectangle $OABC$ is constructed on the Cartesian plane according to these numbers. For this, starting from the point $O$ the points $A_1,A_2,\ldots,A_n$ are consistently marked along the axis $Ox$, and the points $C_1,C_2,\ldots,C_n$ are consistently marked along the axis $Oy$ in such a way that for all $k$ from $1$ to $n$ the lengths of the segments $A_{k-1}A_k=x_k$ and $C_{k-1}C_k=y_k$ are positive integers (let $A_0=C_0=O$, $A_n=A$, and $C_n=C$) and $x_k\cdot y_k=S_k$. [b]a)[/b] Find the maximal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this maximal area is achieved. [b]b)[/b] Find the minimal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this minimal area is achieved. [i](E. Manzhulina, B. Rublyov)[/i]