Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

Let $m,n\geq 2$. One needs to cover the table $m \times n$ using only the following tiles: Tile 1 - A square $2 \times 2$. Tile 2 - A L-shaped tile with five cells, in other words, the square $3 \times 3$ [b]without[/b] the upper right square $2 \times 2$. Each tile 1 covers exactly $4$ cells and each tile 2 covers exactly $5$ cells. Rotation is allowed. Determine all pairs $(m,n)$, such that the covering is possible.

Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.

Let \( X \) be an arbitrary point on the side \( BC \) of triangle \( ABC \). The point \( M \) on the ray \( AB^\to \) beyond \( B \), the point \( N \) on the ray \( AC^\to \) beyond \( C \), and the point \( K \) inside \( ABC \) are such that \( \angle BMX = \angle CNX = \angle KBC = \angle KCB \). The line through \( A \), parallel to \( BC \), intersects the line \( KX \) at point \( P \). Prove that the points \( A \), \( P \), \( M \), \( N \) lie on a circle.

Let $X$ be a point inside convex quadrilateral $ABCD$ with $\angle AXB+\angle CXD=180^{\circ}$. If $AX=14$, $BX=11$, $CX=5$, $DX=10$, and $AB=CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$. [i]Proposed by Michael Kural[/i]

In acute triangle $ABC$ ($AB \neq AC$), $I$ is its incenter and $J$ is the $A$-excenter. $X, Y$ are on minor arcs $\widehat{AB}$ and $\widehat{AC}$ respectively such that $\angle{AXI}=\angle{AYJ}=90^{\circ}$. $K$ is on line $BC$ such that $KI=KJ$. Proof that line $AK$ bisects $\overline{XY}$.

The points $A, B, C, D$ lie in this order on a circle $o$. The point $S$ lies inside $o$ and has properties $\angle SAD=\angle SCB$ and $\angle SDA= \angle SBC$. Line which in which angle bisector of $\angle ASB$ in included cut the circle in points $P$ and $Q$. Prove $PS =QS$.

A [i]simple polygon[/i] is a polygon whose perimeter does not self-intersect. Suppose a simple polygon $\mathcal P$ can be tiled with a finite number of parallelograms. Prove that regardless of the tiling, the sum of the areas of all rectangles in the tiling is fixed.\\ [i]Note:[/i] Points will be awarded depending on the generality of the polygons for which the result is proven.

As shown in the figure, $BQ$ is a diameter of the circumcircle of $ABC$, and $D$ is the midpoint of arc $BC$ (excluding point $A$) . The bisector of the exterior angle of $\angle BAC$ intersects and the extension of $BC$ at point $E$. The ray $EQ$ intersects $\odot (ABC)$ at point $P$. Point $S$ lies on $PQ$ so that $SA = SP$. Point $T$ lies on $BC$ such that $TB = TD$. Prove that $TS \perp SE$. [img]https://cdn.artofproblemsolving.com/attachments/c/4/01460565e70b32b29cddb65d92e041bea40b25.png[/img]

Find the smallest possible area of a convex pentagon whose vertexes are lattice points in a plane.

Let $LEX INGT_1ONMAT_2H$ be a regular $13$-gon. Find $\angle LMT_1$, in degrees. [i]Proposed by Edwin Zhao[/i]

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.

Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ [i] new points [/i], dividing the circle into $ n $ [i] new arcs[/i]. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.) [i]I. Mitrophanov[/i]

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle DAC$, and let $M$ and $N$ be points on segments $BD$ and $CD$, respectively, such that $\angle MAD = \angle DAN$. Let $S, P$ and $Q$ (all different from $A$) be the intersections of the rays $AD$, $AM$ and $AN$ with $\Gamma$, respectively. Show that the intersection of $SM$ and $QD$ lies on $\Gamma$.

Let $O$ be the circumcenter of the triangle $ABC$. Points $D,E$ are on sides $AC,AB$ and points $P,Q,R,S$ are given in plane such that $P,C$ and $R,C$ are on different sides of $AB$ and pints $Q,B$ and $S,B$ are on different sides of $AC$ such that $R,S$ lie on circumcircle of $DAP,EAQ$ and $\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP$(In that order), $\angle ARE=\angle ASD=\angle BAC$, If $RS\| PQ$ prove that $RE ,DS$ are concurrent on $AO$. [i]Proposed by Alireza Dadgarnia[/i]

An incircle of triangle $ABC$ touches $AB$, $BC$, $AC$ at points $C_1$, $A_1$,$ B_1$ respectively. Let $A'$ be the reflection of $A_1$ about $B_1C_1$, point $C'$ is defined similarly. Lines $A'C_1$ and $C'A_1$ meet at point $D$. Prove that $BD \parallel AC$.

2. Find the minimum area of quadrilateral satisfy two condition as follows, (a) At least two interior angles are right angles. (b) A circle radius of $1$ inscribed.

A cat and mouse live on a house mapped out by the points $(-1, 0)$, $(-1, 2)$, $(0, 3)$, $(1, 2)$, $(1, 0)$. The cat starts at the top of the house (point $(0, 3)$) and the mouse starts at the origin (0, 0). Both start running clockwise around the house at the same time. If the cat runs at $12$ units a minute and the mouse at 9 units a minute, how many laps around the house will the cat run before it catches the mouse?

Let $OFT$ and $NOT$ be two similar triangles (with the same orientation) and let $FANO$ be a parallelogram. Show that \[\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.\]

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Let $ABC$ be an arbitrary triangle. Let the excircle tangent to side $a$ be tangent to lines $AB,BC$ and $CA$ at points $C_a,A_a,$ and $B_a,$ respectively. Similarly, let the excircle tangent to side $b$ be tangent to lines $AB,BC,$ and $CA$ at points $C_b,A_b,$ and $B_b,$ respectively. Finally, let the excircle tangent to side $c$ be tangent to lines $AB,BC,$ and $CA$ at points $C_c,A_c,$ and $B_c,$ respectively. Let $A'$ be the intersection of lines $A_bC_b$ and $A_cB_c.$ Similarly, let $B'$ be the intersection of lines $B_aC_a$ and $A_cB_c,$ and let $C$ be the intersection of lines $B_aC_a$ and $A_bC_b.$ Finally, let the incircle be tangent to sides $a,b,$ and $c$ at points $T_a,T_b,$ and $T_c,$ respectively. a) Prove that lines $A'A_a,B'B_b,$ and $C'C_c$ are concurrent. b) Prove that lines $A'T_a, B'T_b,$ and $C'T_c$ are also concurrent, and their point of intersection is on the line defined by the orthocentre and the incentre of triangle $ABC.$ [i]Proposed by Viktor Csaplár, Bátorkeszi and Dániel Hegedűs, Gyöngyös[/i]

Let $\Delta ABC$ be a triangle with circumcenter $O$ and circumcircle $w$. Let $S$ be the center of the circle which is tangent with $AB$, $AC$, and $w$ (in the inside), and let the circle meet $w$ at point $K$. Let the circle with diameter $AS$ meet $w$ at $T$. If $M$ is the midpoint of $BC$, show that $K,T,M,O$ are concyclic.

Diagonal $ DB$ of rectangle $ ABCD$ is divided into $ 3$ segments of length $ 1$ by parallel lines $ L$ and $ L'$ that pass through $ A$ and $ C$ and are perpendicular to $ DB$. The area of $ ABCD$, rounded to the nearest tenth, is [asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B,6)^^rightanglemark(N1,E,B,6)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label("A", A, NE); label("B", B, NE); label("C", C, dir(0)); label("D", D, dir(180)); label("$L$", (x/2,0), SW); label("$L^\prime$", C, SW); label("1", D--F, NW); label("1", F--E, SE); label("1", E--B, SE); clip(W--X--Y--Z--cycle); [/asy] $ \textbf{(A)}\ 4.1 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.3 \qquad \textbf{(D)}\ 4.4 \qquad \textbf{(E)}\ 4.5$

Consider the isosceles triangle $ABC$ with $AB = AC$, and $M$ the midpoint of $BC$. Find the locus of the points $P$ interior to the triangle, for which $\angle BPM+\angle CPA = \pi$.

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]