Two intersecting circles $\omega_1$ and $\omega_2$ are given.Lines $AB,CD$ are common tangents of $\omega_1,\omega_2$($A,C \in \omega_1 ,B,D \in \omega_2$) Let $M$ be the midpoint of $AB$.Tangents through $M$ to $\omega_1$ and $\omega_2$(other than $AB$) intersect $CD$ at $X,Y$.Let $I$ be the incenter of $MXY$.Prove that $IC=ID$.

An acute triangle $ABC$ has $BC$ as its longest side. Points $D,E$ respectively lie on $AC,AB$ such that $BA = BD$ and $CA = CE$. The point $A'$ is the reflection of $A$ against line $BC$. Prove that the circumcircles of $ABC$ and $A'DE$ have the same radii.

Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary. [asy] defaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1); draw(unitcircle,dg); for(int i = 0; i < 12; ++i) { draw(dir(30*i+theta)--dir(30*(i+1)+theta), db); dot(dir(30*i+theta),Fill(rgb(0.8,0,0))); } dot(dir(theta),Fill(dr)); dot((0,0),Fill(dr)); [/asy]

Let $O$ be the circumcenter of triangle $ABC$. Arbitrary points $M$ and $N$ lie on the sides $AC$ and $BC$, respectively. Points $P$ and $Q$ lie in the same half-plane as point $C$ with respect to the line $MN$, and satisfy $\triangle CMN \sim \triangle PAN \sim \triangle QMB$ (in this exact order). Prove that $OP=OQ$. [i]Proposed by Medeubek Kungozhin, Kazakhstan[/i]

The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.

In triangle $ABC$, on line $CA$ it is given point $D$ such that $CD = 3 \cdot CA$ (point $A$ is between points $C$ and $D$), and on line $BC$ it is given point $E$ ($E \neq B$) such that $CE=BC$. If $BD=AE$, prove that $\angle BAC= 90^{\circ}$

In a plane $5$ points are given such that all triangles having vertices at these points are of area not greater than $1$. Show that there exists a trapezoid which contains all point in the interior (or on the sides) and having the area not exceeding $3$.

Two distinct similar rhombi share a diagonal. The smaller rhombus has area $1$, and the larger rhombus has area $9$. Compute the side length of the larger rhombus.

In a given pentagon $ABCDE$, triangles $ABC$, $BCD$, $CDE$, $DEA$ and $EAB$ all have the same area. The lines $AC$ and $AD$ intersect $BE$ at points $M$ and $N$. Prove that $BM = EN$.

Triangle $ABC$ with $AB = 10$ cm ¸and $\angle C= 15^o$, is right at $B$. Point $D \in (AC)$ is the foot of the altitude taken from $B$. Find the distance from point $D$ to the line $AB$.

Let $ ABCD $ be a quadrilateral inscribed in a circle of diameter $ AC. $ Fix points $ E,F $ of segments $ CD, $ respectively, $ BC $ such that $ AE $ is perpendicular to $ DF $ and $ AF $ is perpendicular to $ BE. $ Show that $ AB=AD. $

A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]

\(ABCD\) is a inscribed quadrilateral. \(P\) is the intersection point of its diagonals. \(O\) is its circumcenter. \(\Gamma\) is the circumcircle of \(ABO\). \(\Delta\) is the circumcircle of \(CDO\). \(M\) is the midpoint of arc \(AB\) on \(\Gamma\) who doesn't contain \(O\). \(N\) is the midpoint of arc \(CD\) on \(\Delta\) who doesn't contain \(O\). Show that \(M,N,P\) are collinear.

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles. [asy] unitsize(1.5 cm); pair A, B, C, D, G, H; A = (0,0); B = (2,0); D = (0.5,1.5); C = B + D - A; G = reflect(A,B)*(C) + C - B; H = reflect(B,C)*(H) + A - B; draw(H--A--D--C--G); draw(interp(A,G,-0.1)--interp(A,G,1.1)); draw(interp(C,H,-0.1)--interp(C,H,1.1)); draw(D--G--H--cycle, dashed); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, E); dot("$D$", D, NW); dot("$G$", G, NE); dot("$H$", H, SE); [/asy]

A chord is drawn on a circle by choosing two points uniformly at random along its circumference. This is done two more times to obtain three total random chords. The circle is cut along these three lines, splitting it into pieces. The probability that one of the pieces is a triangle is $\frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m,n) = 1$. Find $100m + n$.

Point $D$ lies on side $BC$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC$. The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F$, respectively. Given that $AB=4$, $BC=5$, $CA=6$, the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt n}p$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

[u]Round 1[/u] [b]p1.[/b] Three two-digit numbers are written on a board. One starts with $5$, another with $6$, and the last one with $7$. Annie added the first and the second numbers; Benny added the second and the third numbers; Denny added the third and the first numbers. Could it be that one of these sums is equal to $148$, and the two other sums are three-digit numbers that both start with $12$? [b]p2.[/b] Three rocks, three seashells, and one pearl are placed in identical boxes on a circular plate in the order shown. The lids of the boxes are then closed, and the plate is secretly rotated. You can open one box at a time. What is the smallest number of boxes you need to open to know where the pearl is, no matter how the plate was rotated? [img]https://cdn.artofproblemsolving.com/attachments/0/2/6bb3a2a27f417a84ab9a64100b90b8768f7978.png[/img] [b]p3.[/b] Two detectives, Holmes and Watson, are hunting the thief Raffles in a library, which has the floorplan exactly as shown in the diagram. Holmes and Watson start from the center room marked $D$. Show that no matter where Raffles is or how he moves, Holmes and Watson can find him. Holmes and Watson do not need to stay together. A detective sees Raffles only if they are in the same room. A detective cannot stand in a doorway to see two rooms at the same time. [img]https://cdn.artofproblemsolving.com/attachments/c/1/6812f615e60a36aea922f145a1ffc470d0f1bc.png[/img] [b]p4.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/4/6/bf0185e142cd3f653d4a9c0882d818c55c64e4.png[/img] [b]p5.[/b] The numbers $1–14$ are placed around a circle in some order. You can swap two neighbors if they differ by more than $1$. Is it always possible to rearrange the numbers using swaps so they are ordered clockwise from $1$ to $14$? [u]Round 2[/u] [b]p6.[/b] A triangulation of a regular polygon is a way of drawing line segments between its vertices so that no two segments cross, and the interior of the polygon is divided into triangles. A flip move erases a line segment between two triangles, creating a quadrilateral, and replaces it with the opposite diagonal through that quadrilateral. This results in a new triangulation. [img]https://cdn.artofproblemsolving.com/attachments/a/a/657a7cf2382bab4d03046075c6e128374c72d4.png[/img] Given any two triangulations of a polygon, is it always possible to find a sequence of flip moves that transforms the first one into the second one? [img]https://cdn.artofproblemsolving.com/attachments/0/9/d09a3be9a01610ffc85010d2ac2f5b93fab46a.png[/img] [b]p7.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9,..., 121)$ are in one column? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Can you draw some diagonals in a convex $2014$-gon so that they do not intersect, the whole $2014$-gon is divided into triangles and each vertex belongs to an odd number of these triangles?

There is convex $n-$gon. We color all its sides and also diagonals, that goes out from one vertex. So we have $2n-3$ colored segments. We write positive numbers on colored segments. In one move we can take quadrilateral $ABCD$ such, that $AC$ and all sides are colored, then remove $AC$ and color $BD$ with number $\frac{xz+yt}{w}$, where $x,y,z,t,w$ - numbers on $AB,BC,CD,DA,AC$. After some moves we found that all colored segments are same that was at beginning. Prove, that they have same number that was at beginning.

The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.

Four regular hexagons surround a square with a side length $1$, each one sharing an edge with the square, as shown in the figure below. The area of the resulting 12-sided outer nonconvex polygon can be written as $m\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$? [asy] import geometry; unitsize(3cm); draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle); draw(shift((1/2,1-sqrt(3)/2))*polygon(6)); draw(shift((1/2,sqrt(3)/2))*polygon(6)); draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6)); draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*polygon(6)); draw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2)); [/asy] $\textbf{(A)}-12~\textbf{(B)}-4~\textbf{(C)} 4~\textbf{(D)}24~\textbf{(E)}32$

Let $ABC$ be a isosceles triangle with $ AC = BC > AB$. Let $ E, F $ be the midpoints of segments $ AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $ l $ meets $ AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $ l$ at $W$. Let $ P $ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $ AW = PW $ if and only if $B$ lies on the circumcircle of $EFM$.

Consider two spheres $\Sigma_1$ and $\Sigma_2$ and a line $\Delta$ not meeting them. Let $C_i$ and $r_i$ be the center and radius of $\Sigma_i$, and let $H_i$ and $d_i$ be the orthogonal projection of $C_i$ onto $\Delta$ and the distance of $C_i$ from $\Delta~(i=1,2)$. For a point $M$ on $\Delta$, let $\delta_i(M)$ be the length of a tangent $MT_i$ to $\Sigma_i$, where $T_i\in\Sigma_i~(i=1,2)$. Find $M$ on $\Delta$ for which $\delta_1(M)+\delta_2(M)$ is minimal.

Let $ABC$ be an arbitrary triangle. A circle passes through $B$ and $C$ and intersects the lines $AB$ and $AC$ at $D$ and $E$, respectively. The projections of the points $B$ and $E$ on $CD$ are denoted by $B'$ and $E'$, respectively. The projections of the points $D$ and $C$ on $BE$ are denoted by $D'$ and $C'$, respectively. Prove that the points $B',D',E'$ and $C'$ lie on the same circle.

Let $ABCDE$ be a cyclic pentagon such that $BC = DE$ and $AB$ is parallel to $DE$. Let $X, Y,$ and $Z$ be the midpoints of $BD, CE,$ and $AE$ respectively. Show that $AE$ is tangent to the circumcircle of the triangle $XYZ$. Proposed by [i]Nikola Velov, Macedonia[/i]