Let $\Gamma$ be a circunference and $O$ its center. $AE$ is a diameter of $\Gamma$ and $B$ the midpoint of one of the arcs $AE$ of $\Gamma$. The point $D \ne E$ in on the segment $OE$. The point $C$ is such that the quadrilateral $ABCD$ is a parallelogram, with $AB$ parallel to $CD$ and $BC$ parallel to $AD$. The lines $EB$ and $CD$ meets at point $F$. The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at $I$. Prove that the line $EI$ is the angle bissector of $\angle BEC$.

[color=darkblue]Let $ ABCD$ be a trapezoid with $ AB\parallel CD$. Exterior equilateral triangles $ ABE$ and $ CDF$ are constructed. Prove that lines $ AC$, $ BD$ and $ EF$ are concurrent.[/color]

A straight line cuts the side $AB$ of the triangle $ABC$ at $C_1$, the side $AC$ at $B_1$ and the line $BC$ at $A_1$. $C_2$ is the reflection of $C_1$ in the midpoint of $AB$, and $B_2$ is the reflection of $B_1$ in the midpoint of $AC$. The lines $B_2C_2$ and $BC$ intersect at $A_2$. Prove that $$\frac{sen \, \, B_1A_1C}{sen\, \, C_2A_2B} = \frac{B_2C_2}{B_1C_1}$$ [img]https://cdn.artofproblemsolving.com/attachments/3/8/774da81495df0a0f7f2f660ae9f516cf70df06.png[/img]

$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.

Given is triangle $ABC$ with centroid $T$, such that $\angle BAC + \angle BTC = 180^\circ$. Let $G$ and $H$ be the second points of intersection of lines $CT$ and $BT$ with the circumcircle of triangle $ABC$, respectively. Prove that the line $GH$ is tangent to the Euler circle of triangle $ABC$. [i]Proposed by Andrija Živadinović[/i]

A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD = ED.$

Circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively lie on a plane such that that the circle $C_2$ passes through $O_1$. The ratio of radius of circle $C_1$ to $O_1O_2$ is $\sqrt{2+\sqrt3}$. a) Prove that the circles $C_1$ and $C_2$ intersect at two distinct points. b) Let $A,B$ be these points of intersection. What proportion of the area of circle is $C_1$ is the area of the sector $AO_1B$ ?

Determine the equations of a surface in three-dimensional cartesian space which has the following properties: (a) it passes through the point $(1,1,1)$ and (b) if the tangent plane is drawn at any point $P$ and $X,Y, Z$ are the intersections of this plane with the $x, y$ and $z-$axis respectively, then $P$ is the orthocenter of the triangle $XYZ.$

Let $\triangle ABC$ satisfy $AB = 17, AC = \frac{70}{3}$ and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a, b$ are coprime positive integers, find $a + b$.

Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles? Can a square be dissected into $15$ congruent polygons which are not rectangles?

In rectangle $ABCD$, $AB = 3$ and $BC = 4$. If the feet of the perpendiculars from $B$ and $D$ to $AC$ are $X$ and $Y$ , the length of $X Y$ can be expressed in the form m/n , where m and n are relatively prime positive integers. Find $m +n$.

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

Quadrilateral $ABCD$ is inscribed into circle $\omega$, $AC$ intersect $BD$ in point $K$. Points $M_1$, $M_2$, $M_3$, $M_4$-midpoints of arcs $AB$, $BC$, $CD$, and $DA$ respectively. Points $I_1$, $I_2$, $I_3$, $I_4$-incenters of triangles $ABK$, $BCK$, $CDK$, and $DAK$ respectively. Prove that lines $M_1I_1$, $M_2I_2$, $M_3I_3$, and $M_4I_4$ all intersect in one point.

$ABC$ is an acute isosceles triangle $(AB=AC)$ and $CD$ one altitude. Circle $C_2(C,CD)$ meets $AC$ at $K$, $AC$ produced at $Z$ and circle $C_1(B, BD)$ at $E$. $DZ$ meets circle $(C_1)$ at $M$. Show that: a) $\widehat{ZDE}=45^0$ b) Points $E, M, K$ lie on a line. c) $BM//EC$

Let $ \triangle ABC$ be a triangle and let $ K$ be some point on the side $ AB$, so that the tangent line from $ K$ to the incircle of $ \triangle ABC$ intersects the ray $ AC$ at $ L$. Assume that $ \omega$ is tangent to sides $ AB$ and $ AC$, and to the circumcircle of $ \triangle AKL$. Prove that $ \omega$ is tangent to the circumcircle of $ \triangle ABC$ as well.

Two circunmferences $ \Gamma_1$ $ \Gamma_2$ intersect at $ A$ and $ B$ $ r_1$ is the tangent from $ A$ to $ \Gamma_1$ and $ r_2$ is the tangent from $ B$ to $ \Gamma_2$ $ r_1 \cap r_2\equal{}C$ $ T\equal{} r_1 \cap \Gamma_2$ ($ T \neq A$) We consider a point $ X$ in $ \Gamma_1$ which is distinct from $ A$ and $ B$. $ XA \cap \Gamma_2 \equal{}Y$ ($ Y \neq A$) $ YB \cap XC\equal{}Z$ Prove that $ TZ \parallel XY$

Let $ APD $ be an acute-angled triangle and let $ B,C $ be two points on the segments (excluding their endpoints) $ AP,PD, $ respectively. The diagonals of $ ABCD $ meet at $ Q. $ Denote by $ H_1,H_2 $ the orthocenters of $ APD,BPC, $ respectively. The circumcircles of $ ABQ $ and $ CDQ $ intersect at $ X\neq Q, $ and the circumcircles of $ ADQ,BCQ $ meet at $ Y\neq Q. $ Prove that if the line $ H_1H_2 $ passes through $ X, $ then it also passes through $ Y. $

Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$

Let $ ABC$ be an acute-angled triangle such that there exist points $ D,E,F$ on side $ BC,CA,AB$, respectively such that the inradii of triangle $ AEF,BDF,CDE$ are all equal to $ r_0$. If the inradii of triangle $ DEF$ and $ ABC$ are $ r$ and $ R$, respectively, prove that \[ r\plus{}r_0\equal{}R.\] [i]Soewono, Bandung[/i]

A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.

A sphere with unit radius is given. Furthermore, circles $k_0,k_1,\ldots,k_n\ (n\ge3)$ of the same radius $r$ are given on the sphere. The circle $k_0$ is tangent to all other circles $k_i$ and every two circles $k_i,k_{i+1}$ are tangent for $i=1,\ldots,n$ (assuming $k_{n+1}=k_1$). a) Find relation between numbers $n,r.$ b) Determine for which $n$ the described situation can occur and compute the corresponding radius $r.$ (We say non-planar circles are tangent if they have only a single common point and their tangent lines in this point coincide.)

Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$. [i]Original formulation:[/i] Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$

Let $P$ be a point outside a circle centered at $O$. From $P$, tangent lines are drawn to the circle, touching the circle at points $A$ and $B$. Ray $\overrightarrow{BO}$ is drawn intersecting the circle again at $C$ and intersecting ray $\overrightarrow{PA}$ at $Q$. If $3QA = 2AP$, what is the value of $\sin \angle CAQ$?

An ant is at one corner of a unit cube. If the ant must travel on the box’s surface, the shortest distance the ant must crawl to reach the opposite corner of the cube can be written in the form $\sqrt{a}$, where $a$ is a positive integer. Compute $a$.