As shown in the figure, two circles $\Gamma_1$ and $\Gamma_2$ on the plane intersect at two points $A$ and $B$. The two rays passing through $A$, $\ell_1$ and $\ell_2$ intersect $\Gamma_1$ at points $D$ and $E$ respectively, and $\Gamma_2$ at points $F$ and $C$ respectively (where $E$ and $F$ lie on line segments $AC$ and $AD$ respectively, and neither of them coincides with the endpoints). It is known that the three lines $AB$, $CF$ and $DE$ have a common point, the circumscribed circle of $\vartriangle AEF$ intersects $AB$ at point $G$, the straight line $EG$ intersects the circle $\Gamma_1$ at point $P$, the straight line $FG$ intersects the circle $\Gamma_2$ at point $Q$. Let the symmetric points of $C$ and $D$ wrt the straight line $AB$ be $C'$ and $D'$ respectively. If $PD'$ and $QC'$ intersect at point$ J$, prove that $J$ lies on the straight line $AB$. [img]https://cdn.artofproblemsolving.com/attachments/3/7/eb3acdbad52750a6879b4b6955dfdb7de19ed3.png[/img]

In rectangle $ABCD$, $AB=6$, $AD=30$, and $G$ is the midpoint of $\overline{AD}$. Segment $AB$ is extended $2$ units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{ED}$ and $\overline{BC}$. What is the area of $BFDG$? $ \textbf{(A)}\ \frac{133}{2}\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ \frac{135}{2}\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ \frac{137}{2}$

Given triangle $ABC$. Point $A_1$ is symmetric to vertex $A$ wrt line $BC$, and point $C_1$ is symmetric to vertex $C$ wrt line $AB$. Prove that if points $A_1$, $B$ and $C_1$ lie on the same line and $C_1B = 2A_1B$, then angle $\angle CA_1B$ is right.

Some $ m$ squares on the chessboard are marked. If among four squares at the intersection of some two rows and two columns three squares are marked then it is allowed to mark the fourth square. Find the smallest $ m$ for which it is possible to mark all squares after several such operations.

Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions: $(1)$Square bar $AB$ is bisected by square bar $CD$ $(2)$At least one square of $AB$ lay on both sides of $CD$ $(3)$At least one square of $CD$ lay on both sides of $AB$ There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet

Let $A_1$ and $C_1$ be the tangency points of the incircle of triangle $ABC$ with $BC$ and $AB$ respectively, $A'$ and $C'$ be the tangency points of the excircle inscribed into the angle $B$ with the extensions of $BC$ and $AB$ respectively. Prove that the orthocenter $H$ of triangle $ABC$ lies on $A_1C_1$ if and only if the lines $A'C_1$ and $BA$ are orthogonal.

Find all points inside a given equilateral triangle such that the distances from it to three sides of the given triangle are the side lengths of a triangle.

Let $ABCD$ be a rectangle. Two perpendicular lines pass through point $B$. One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$. Let $F$ be the common point of $KL$ and $AC$. Prove that $BF\perp KL$.

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Let $\vartriangle ABC$ be an isosceles triangle with $AC = BC$. Points $D, E, F$ are considered on $BC, CA, AB$, respectively, such that $AF> BF$ and that the quadrilateral $CEFD$ is a parallelogram. The perpendicular line to $BC$ drawn by $B$ intersects the perpendicular bisector of $AB$ at $G$. Prove that $DE \perp FG$.

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Point $D$ lies on arc $\frown{BC}$ not containing $A$. Let $E$ be the foot of perpendicular from $A$ to line $CD$. Prove that $BC+DC=2DE$.

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.

Six small circles of radius $1$ are drawn so that they are all tangent to a larger circle, and two of them are tangent to sides $BC$ and $AD$ of a rectangle $ABCD$ at their midpoints $K$ and $L$. Each of the remaining four small circles is tangent to two sides of the rectangle. The large circle is tangent to sides $AB$ and $CD$ of the rectangle and cuts the other two sides. Find the radius of the large circle. [img]https://cdn.artofproblemsolving.com/attachments/b/4/a134da78d709fe7162c48d6b5c40bd1016c355.png[/img]

Let $ABC$ be a triangle in which the length of side $AB$ is $4$ units, and that of $BC$ is $2$ units. Let $D$ be the point on $AB$ at distance $3$ units from $A$. Prove that the line perpendicular to $AB$ through $D$, the angle bisector of $\angle ABC$, and the perpendicular bisector of $BC$ all meet at a single point.

$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes. $ (b)$ Prove that every integer can be written as a sum of five cubes.

Let $ABC$ be a triangle inscribed in circle $(O)$ with diamter $KL$ passes through the midpoint $M$ of $AB$ such that $L, C$ lie on the different sides respect to $AB$. A circle passes through $M, K $cuts $LC$ at$ P, Q $(point $P$ lies between$ Q, C$). The line $KQ $cuts $(LMQ)$ at $R$. Prove that $ARBP$ is cyclic and$ AB$ is the symmedian of triangle $APR$. Please help :)

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)

Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.

Let $ABC$ be an acute scalene triangle, $H$ its orthocenter and $G$ its geocenter. The circumference with diameter $AH$ cuts the circumcircle of $BHC$ in $A'$ ($A' \neq H$). Points $B'$ and $C'$ are defined similarly. Show that the points $A'$, $B'$, $C'$, and $G$ lie in one circumference.

Let $ABC$ be a non-degenerate triangle with incenter $I$ and circumcircle $\Gamma$. Denote by $M_a$ the midpoint of the arc $\widehat{BC}$ of $\Gamma$ not containing $A$, and define $M_b$, $M_c$ similarly. Suppose $\triangle ABC$ has inradius $4$ and circumradius $9$. Compute the maximum possible value of \[IM_a^2+IM_b^2+IM_c^2.\][i]Proposed by David Altizio[/i]

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

The figure on the right shows the ellipse $\frac{(x-19)^2}{19}+\frac{(x-98)^2}{98}=1998$. Let $R_1,R_2,R_3,$ and $R_4$ denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of $R_1-R_2+R_3-R_4$. [asy] defaultpen(linewidth(0.7)); pair c=(19,98); real dist = 30; real a = sqrt(1998*19),b=sqrt(1998*98); xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow); yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow); draw(ellipse(c,a,b)); label("$R_1$",(100,200)); label("$R_2$",(-80,200)); label("$R_3$",(-60,-150)); label("$R_4$",(70,-150));[/asy]

Consider a fixed triangle $ABC$ such that $AB=AC.$ Let $M$ be the midpoint of $BC.$ Let $P$ be a variable point inside $\triangle ABC,$ such that $\angle PBC=\angle PCA.$ Prove that the sum of the measures of $\angle BPM$ and $\angle APC$ is constant.

Let $I$ be the incenter of a nonisosceles triangle $ABC$. Prove that there exists a unique pair of points $M$, $N$ lying on the sides $AC$, $BC$ respectively, such that $\angle AIM = \angle BIN$ and $MN|| AB$.