Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$. Let $K$ be the midpoint of $BC$, and let $AKLM$ be a parallelogram with centre $C$. Let $T$ be the intersection of the line $AC$ and the perpendicular bisector of $BM$. Let $\omega_1$ be the circle with centre $C$ and radius $CA$ and let $\omega_2$ be the circle with centre $T$ and radius $TB$. Prove that one of the points of intersection of $\omega_1$ and $\omega_2$ is on the line $LM$. [i]Proposed by Greece[/i]

In a scalene triangle $ABC$,the bisectors of angle $A,B$ intersect their corresponding sides at $K,L$ respectively.$I,O,H$ denote respectively the incenter,circumcenter and orthocenter of triangle $ABC$. Prove that $A,B,K,L,O$ are concyclic iff $KL$ is the common tangent line of the circumcircles of the three triangles $ALI,BHI$ and $BKI$.

Let $\triangle ABC$ be a triangle with sides $a,b,c$ and corresponding angles $A,B,C$ (so $a=BC$ and $A=\angle BAC$ etc.). Suppose that $4A+3C=540^\circ$. Prove that $(a-b)^2(a+b)=bc^2$.

In the triangle $ABC$ it is known that $2AC=AB$ and $\angle A = 2\angle B$. In this triangle draw the angle bisector $AL$, and mark point $M$, the midpoint of the side $AB$. It turned out that $CL = ML$. Prove that $\angle B= 30^o$. (Hilko Danilo)

Let $ABC$ be an acute-angled triangle with $AC>BC$ and circumcircle $\omega$. Suppose that $P$ is a point on $\omega$ such that $AP=AC$ and that $P$ is an interior point on the shorter arc $BC$ of $\omega$. Let $Q$ be the intersection point of the lines $AP$ and $BC$. Furthermore, suppose that $R$ is a point on $\omega$ such that $QA=QR$ and $R$ is an interior point of the shorter arc $AC$ of $\omega$. Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$. Prove that the points $P, Q, R$ and $S$ are concyclic. [i]Proposed by Patrik Bak, Slovakia[/i]

Let there be given two lines $D_1$ and $D_2$ which intersect at point $O$, and a point $M$ not on any of these lines. Consider two variable points $A\in D_1$ and $b\in D_2$ such that $M$ belongs to the segment $AB$. (a) Prove that there exists a position of $A$ and $B$ for which the area of triangle $OAB$ is minimal. Construct such points $A$ and $B$. (b) Prove that there exists a position of $A$ and $B$ for which the area of triangle $OAB$ is minimal. Show that for such $A$ and $B$, the perimeters of $\triangle OAM$ and $\triangle OBM$ are equal, and that $\frac{AM}{\tan\frac12\angle OAM}=\frac{BM}{\tan\frac12\angle OBM}$. Construct such points $A$ and $B$.

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

Let $P$ and $Q$ be distinct points in the plane of a triangle $ABC$ such that $AP : AQ = BP : BQ = CP : CQ$. Prove that the line $PQ$ passes through the circumcenter of the triangle.

In an acute triangle $ABC$ , the bisector of angle $\angle A$ intersects the circumscribed circle of the triangle $ABC$ at the point $W$. From point $W$ , a parallel is drawn to the side $AB$, which intersects this circle at the point $F \ne W$. Describe the construction of the triangle $ABC$, if given are the segments $FA$ , $FW$ and $\angle FAC$. (Andrey Mostovy)

Let $S$ be a circle with center $C$ and radius $r$, and let $P \ne C$ be an arbitrary point. A line $\ell$ through $P$ intersects the circle in $X$ and $Y$. Let $Z$ be the midpoint of $XY$. Prove that the points $Z$, as $\ell$ varies, describe a circle. Find the center and radius of this circle.

Square $ABCD$ with side length $2$ begins in position $1$ with side $AD$ horizontal and vertex $A$ in the lower right corner. The square is rotated $90^o$ clockwise about vertex $ A$ into position $2$ so that vertex $D$ ends up where vertex $B$ was in position $1$. Then the square is rotated $90^o$ clockwise about vertex $C$ into position $3$ so that vertex $B$ ends up where vertex $D$ was in position $2$ and vertex $B$ was in position $1$, as shown below. The area of the region of points in the plane that were covered by the square at some time during its rotations can be written $\frac{p\pi + \sqrt{q} + r}{s}$, where $p, q, r,$ and $s$ are positive integers, and $p$ and $s$ are relatively prime. Find $p + q + r + s$. [img]https://cdn.artofproblemsolving.com/attachments/9/2/cb15769c30018545abfa82a9f922201c4ae830.png[/img]

Let $\ell$ be a line in the plane. Two circles with respective radii $2$ and $4$ are tangent to $\ell$ on the same side so that their points of tangency are distance $9$ apart. The two common internal tangents to both circles are drawn. What is the area of the triangle formed by the line $\ell$ and the two internal tangents? $\text{(A) }\frac{25}{3}\qquad\text{(B) }\frac{26}{3}\qquad\text{(C) }9\qquad\text{(D) }\frac{28}{3}\qquad\text{(E) }\frac{29}{3}$

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here. Problem 1. Given an acute triangle $ABC$ and the point $D$ on segment $BC$. Circle $c_1$ passes through $A, D$ and its centre lies on $AC$. Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$. Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$. Prove that $AD$ bisects $\angle{PDQ}$.

Consider a convex $15$- gon with perimeter $21$. Show that there one can select three distinct pairs of vertices that form a triangle with area less than $1$. [hide=original wording of second sentence]Zeige, dass man davon drei paarweise verschiedene Eckpunkte auswählen kann, die ein Dreieck mit Fläche kleiner als 1 bilden.[/hide]

Consider the rectangular strip of length $12$ below, divided into three rectangles. The distance between the centers of two of the rectangles is $4$. What is the length of the other rectangle? [asy] size(120); draw((0,0)--(12,0)--(12,1)--(0,1)--cycle); draw((8,1)--(8,0)); draw((3,1)--(3,0)); dot((1.5,0.5)); dot((5.5,0.5)); draw((1.5,0.5)--(5.5,0.5)); [/asy] $\textbf{(A) }2.5\qquad\textbf{(B) }3\qquad\textbf{(C) }3.5\qquad\textbf{(D) }4\qquad\textbf{(E) }4.5$

Let $ABCD$ be a square of side length 2, and let $M$ and $N$ be points on the sides $AB$ and $CD$ respectively. The lines $CM$ and $BN$ meet at $P$, while the lines $AN$ and $DM$ meet at $Q$. Prove that $\left| PQ \right|\ge 1$.

Let $ABC$ be a triangle with $\angle A = 60^\circ$, and $AA', BB', CC'$ be its internal angle bisectors. Prove that $\angle B'A'C' \le 60^\circ$.

$ABC$ is a triangle. Take $a = BC$ etc as usual. Take points $T_1, T_2$ on the side $AB$ so that $AT_1 = T_1T_2 = T_2B$. Similarly, take points $T_3, T_4$ on the side BC so that $BT_3 = T_3T_4 = T_4C$, and points $T_5, T_6$ on the side $CA$ so that $CT_5 = T_5T_6 = T_6A$. Show that if $a' = BT_5, b' = CT_1, c'=AT_3$, then there is a triangle $A'B'C'$ with sides $a', b', c'$ ($a' = B'C$' etc). In the same way we take points $T_i'$ on the sides of $A'B'C' $ and put $a'' = B'T_6', b'' = C'T_2', c'' = A'T_4'$. Show that there is a triangle $A'' B'' C'' $ with sides $a'' b'' , c''$ and that it is similar to $ABC$. Find $a'' /a$.

A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid. [asy] size(150); defaultpen(linewidth(0.8)); draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1)); draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4")); draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4")); label("$5$",(0,5/2),W); label("$8$",(4,0),S); [/asy]

Is it possible to place a triangle with area $1999$ and perimeter $19992$ in the interior of a triangle with area $2000$ and perimeter $20002$?

In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid? [asy] unitsize(2cm); path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle; draw(box); label("$+$",(0,0)); draw(shift(1,0)*box); label("$-$",(1,0)); draw(shift(2,0)*box); label("$+$",(2,0)); draw(shift(3,0)*box); label("$-$",(3,0)); draw(shift(0.5,0.4)*box); label("$-$",(0.5,0.4)); draw(shift(1.5,0.4)*box); label("$-$",(1.5,0.4)); draw(shift(2.5,0.4)*box); label("$-$",(2.5,0.4)); draw(shift(1,0.8)*box); label("$+$",(1,0.8)); draw(shift(2,0.8)*box); label("$+$",(2,0.8)); draw(shift(1.5,1.2)*box); label("$+$",(1.5,1.2)); [/asy] $\textbf{(A) } 2 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16$

A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$? $\textbf{(A) }5\sqrt2+4\qquad\textbf{(B) }\sqrt{17}+7\qquad\textbf{(C) }6\sqrt2+3\qquad\textbf{(D) }\sqrt{15}+8\qquad\textbf{(E) }12$

Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.

The points $A_1,B_1,C_1$ belong to $[BC],[CA],[AB]$ sides of the $ABC$ triangle respectively. The $[AA_1], [BB_1], [CC_1]$ segments split the $ABC$ onto $4$ smaller triangles and $3$ quadrangles. It is known, that the smaller triangles have the same area. Prove that the quadrangles have equal areas. What is the quadrangle area, it the small triangle has the unit area?