In a trapezoid $ABCD$ with bases $AD$ and $BC$, the bisector of the angle $\angle DAB$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $P$ and $S$, respectively, and the bisector of the angle $\angle BCD$ intersects the bisectors of the angles $\angle ABC$ and $\angle CDA$ at the points $Q$ and $R$, respectively. Prove that if $PS\parallel RQ$, then $AB = CD$.

Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.

Let $ABC$ be an acute angled triangle$.$ Let $D$ be the foot of the internal angle bisector of $\angle BAC$ and let $M$ be the midpoint of $AD.$ Let $X$ be a point on segment $BM$ such that $\angle MXA=\angle DAC.$ Prove that $AX$ is perpendicular to $XC.$

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.

The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is a) tangential (circumscribed), b) cyclic (inscribed). (Nairi Sedrakyan)

There is an integer in each cell of a $2m\times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a necessary and sufficient condition, in terms of $m$, $n$ and the initial numbers on the table, for which there exists a sequence of operations that makes all the numbers on the table equal.

In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD =DP = 3$, and $GC = 4$, find $AB^2$. [i]Proposed by Muztaba Syed[/i]

In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.

[b]p1.[/b] Each point in the interior and on the boundary of a square of side $2$ inches is colored either red or blue. Prove that there exists at least one pair of points of the same color whose distance apart is not less than $-\sqrt5$ inches. [b]p2.[/b] $ABC$ is an equilateral triangle of altitude $h$. A circle with center $0$ and radius $h$ is tangent to side $AB$ at $Z$ and intersects side $AC$ in point $X$ and side $BC$ in point $Y$. Prove that the circular arc $XZY$ has measure $60^o$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/ac70942f7a14cd0759ac682c3af3551687dd69.png[/img] [b]p3.[/b] Find all of the real and complex solutions (if any exist) of the equation $x^7 + 7^7 = (x + 7)^7$ [b]p4.[/b] The four points $A, B, C$, and $D$ are not in the same plane. Given that the three angles, angle $ABC$, angle $BCD$, and angle $CDA$, are all right angles, prove that the fourth angle, angle $DAB$, of this skew quadrilateral is acute. [b]p5.[/b] $A, B, C$ and $D$ are four positive whole numbers with the following properties: (i) each is less than the sum of the other three, and (ii) each is a factor of the sum of the other three. Prove that at least two of the numbers must be equal. (An example of four such numbers: $A = 4$, $B = 4$, $C = 2$, $D = 2$.) [b]p6.[/b] $S$ is a set of six points and $L$ is a set of straight line segments connecting certain pairs of points in $S$ so that each point of $S$ is connected with at least four of the other points. Let $A$ and $B$ denote two arbitrary points of $S$. Show that among the triangles having sides in $L$ and vertices in $S$ there are two with the properties: (i) The two triangles have no common vertex. (ii) $A$ is a vertex of one of the triangles, and $B$ is a vertex of the other. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Triangle $ABC$ is drawn such that $\angle A = 80^o$, $\angle B = 60^o$, and $\angle C = 40^o$. Let the circumcenter of $\vartriangle ABC$ be $O$, and let $\omega$ be the circle with diameter $AO$. Circle $\omega$ intersects side $AC$ at point $P$. Let M be the midpoint of side $BC$, and let the intersection of $\omega$ and $PM$ be $K$. Find the measure of $\angle MOK$.

Let $ABC$ be a triangle and let its incircle $\gamma$ touch the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $P$ be a point strictly in the interior of $\gamma.$ The segments $PA,PB,PC$ cross $\gamma$ at $A_0,B_0,C_0$ respectively. Let $S_A,S_B,S_C$ be the centres of the circles $PEF,PFD,PDE$ respectively and let $T_A,T_B,T_C$ be the centres of the circles $PB_0C_0,PC_0A_0,PA_0B_0$ respectively. Prove that $S_AT_A, S_BT_B$ and $S_CT_C$ are concurrent.

Let $ABCDEF$ be a convex hexagon with $AB = CD = EF$, $BC =DE = FA$ and $\angle A+\angle B = \angle C +\angle D = \angle E +\angle F$. Prove that $\angle A=\angle C=\angle E$ and $\angle B=\angle D=\angle F$. Tran Quang Hung

Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.

A rectangle, whose one sidelength is twice the other side, is inscribed inside a triangles with sides $3$ cm, $4$ cm and $5$ cm, such that the long sides lies entirely on the long side of the triangle. The other two remaining vertices of the rectangle lie respectively on the other two sides of the triangle. Find the lengths of the sides of this rectangle.

Let $M, N$ be the midpoints of sides $AB, AC$ respectively of a triangle $ABC$. The perpendicular bisector to the bisectrix $AL$ meets the bisectrixes of angles $B$ and $C$ at points $P$ and $Q$ respectively. Prove that the common point of lines $PM$ and $QN$ lies on the tangent to the circumcircle of $ABC$ at $A$.

Consider a circle and a point $A$ outside it. We start moving from $A$ along a closed broken line consisting of segments of tangents to the circle (the segment itself should not necessarily be tangent to the circle) and terminate back at $A$. (On the links of the broken line are solid.) We label parts of the segments with a plus sign if we approach the circle and with a minus sign otherwise. Prove that the sum of the lengths of the segments of our path, with the signs given, is zero. [img]https://cdn.artofproblemsolving.com/attachments/3/0/8d682813cf7dfc88af9314498b9afcecdf77d2.png[/img]

Let $\omega$ be the incircle of triangle $ABC$ and $\omega$ touches $BC$ at $D$. $AD$ meets $\omega$ again at $L$. Let $K$ be $A$-excenter, and $M,N$ be the midpoint of $BC,KM$, respectively. Prove that $B,C,N,L$ are concyclic.

Three mutually tangent spheres each with radius $5$ sit on a horizontal plane. A triangular pyramid has a base that is an equilateral triangle with side length $6$, has three congruent isosceles triangles for vertical faces, and has height $12$. The base of the pyramid is parallel to the plane, and the vertex of the pyramid is pointing downward so that it is between the base and the plane. Each of the three vertical faces of the pyramid is tangent to one of the spheres at a point on the triangular face along its altitude from the vertex of the pyramid to the side of length $6$. The distance that these points of tangency are from the base of the pyramid is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(200); defaultpen(linewidth(0.8)); pair X=(-.6,.4),A=(-.4,2),B=(-.7,1.85),C=(-1.1,2.05); picture spherex; filldraw(spherex,unitcircle,white); draw(spherex,(-1,0)..(-.2,-.2)..(1,0)^^(0,1)..(-.2,-.2)..(0,-1)); add(shift(-0.5,0.6)*spherex); filldraw(X--A--C--cycle,gray); draw(A--B--C^^X--B); add(shift(-1.5,0.2)*spherex); add(spherex); [/asy]

$ABC$ is a triangle in the plane. Find the locus of point $P$ for which $PA,PB,PC$ form a triangle whose area is equal to one third of the area of triangle $ABC$.

In the plane, the point $M$ is the midpoint of a line segment $[AB]$ and $\ell$ is an arbitrary line that has no has a common point with the line segment $[AB]$ (and is also not perpendicular to $[AB]$). The points $X$ and $Y$ are the perpendicular projections of $A$ and $B$ onto $\ell$, respectively. Show that the circumscribed circles of triangle $\vartriangle AMX$ and triangle $\vartriangle BMY$ have the same radius.

Form a pentagon by taking a square of side length $1$ and an equilateral triangle of side length $1$ and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one vertex of the equilateral triangle, and exactly two vertices of the square. What is the radius of the circle? $\textbf{(A) }\dfrac23\hspace{14.4em}\textbf{(B) }\dfrac34\hspace{14.4em}\textbf{(C) }1$ $\textbf{(D) }\dfrac54\hspace{14.4em}\textbf{(E) }\dfrac43\hspace{14.4em}\textbf{(F) }\dfrac{\sqrt2}2$ $\textbf{(G) }\dfrac{\sqrt3}2\hspace{13.5em}\textbf{(H) }\sqrt2\hspace{13.8em}\textbf{(I) }\sqrt3$ $\textbf{(J) }\dfrac{1+\sqrt3}2\hspace{12em}\textbf{(K) }\dfrac{2+\sqrt6}2\hspace{11.9em}\textbf{(L) }\dfrac76$ $\textbf{(M) }\dfrac{2+\sqrt6}4\hspace{11.5em}\textbf{(N) }\dfrac45\hspace{14.4em}\textbf{(O) }2007$

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

Let $ABCD$ be a convex trapezoid such that $\angle ABC = \angle BCD = 90^o$, $AB = 3$, $BC = 6$, and $CD = 12$. Among all points $X$ inside the trapezoid satisfying $\angle XBC = \angle XDA$, compute the minimum possible value of $CX$.

Prove that if two segments of a tetrahedron, going from the ends of some edge to the centers of the inscribed circles of opposite faces, intersect, then the segments issued from the ends of the crossing with it edges to the centers of the inscribed circles of the other two faces, also intersect.