Let $ABCD$ be a trapezoid with bases $AB$ and $CD$, inscribed in a circle of center $O$. Let $P$ be the intersection of the lines $BC$ and $AD$. A circle through $O$ and $P$ intersects the segments $BC$ and $AD$ at interior points $F$ and $G$, respectively. Show that $BF=DG$.

A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures 8 inches high and 10 inches wide. What is the area of the border, in square inches? $\textbf{(A)}\hspace{.05in}36 \qquad \textbf{(B)}\hspace{.05in}40 \qquad \textbf{(C)}\hspace{.05in}64 \qquad \textbf{(D)}\hspace{.05in}72 \qquad \textbf{(E)}\hspace{.05in}88 $

Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.

Let $T$ be a triangle with inscribed circle $C.$ A square with sides of length $a$ is circumscribed about the same circle $C.$ Show that the total length of the parts of the edge of the square interior to the triangle $T$ is at least $2 \cdot a.$

In the scalene triangle $ABC$, the bisector of angle A cuts side $BC$ at point $D$. The tangent lines to the circumscribed circunferences of triangles $ABD$ and $ACD$ on point D, cut lines $AC$ and $AB$ on points $E$ and $F$ respectively. Let $G$ be the intersection point of lines $BE$ and $CF$. Prove that angles $EDG$ and $ADF$ are equal.

Inside the triangle $ABC$ choose the point $M$, and on the side $BC$ - the point $K$ in such a way that $MK || AB$. The circle passing through the points $M, \, \, K, \, \, C,$ crosses the side $AC$ for the second time at the point $N$, a circle passing through the points $M, \, \, N, \, \, A, $ crosses the side $AB$ for the second time at the point $Q$. Prove that $BM = KQ$. (Nagel Igor)

Let $ABCD$ be a rectangle of center $O$, such that $\angle DAC = 60^o$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$ and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

Prove that the equilateral triangle of area $1$ can be covered by five arbitrary equilateral triangles having the total area $2$.

Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$, and let $M$ be the midpoint of $BC$. Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$.

Let a hexagone with a diameter $D$ be given and let $d>\frac D 2.$ On each side of the hexagon one constructs a isosceles triangle with two equal sides of length $d$. Prove that the sum of the areas of those isoscele triangles is greater than the area of a rhombus with side lengths $d$ and a diagonal of length $D$. (The diameter of a polygon is the maximum of the lengths of all its sides and diagonals.)

Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic. (D.Prokopenko)

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

Let $G$ be the centroid of $\triangle ABC $ and let $D, E $ and $F$ be the midpoints of the line segments $BC, CA $ and $AB$ respectively. Suppose the circumcircle of $\triangle ABC $ meets $AD $ again at $X$, the circumcircle of $\triangle DEF $ meets $BE$ again at $Y$ and the circumcircle of $\triangle DEF $ meets $CF$ again at $Z$. Show that $G, X, Y $ and $Z$ are concyclic.

Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m \plus{} 2)\times(4m \plus{} 3)$ can be covered with $ \dfrac{n(n \plus{} 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n \minus{} 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.

Given two lines in a plane, intersecting at an acute angle. In the direction of one of the straight lines, compression is performed with a coefficient of 1/2. Prove that there is a point from which the distance to the point of intersection of the lines increases. Note: What is meant here is a transformation in which each point moves parallel to one straight line so that its distance to the second straight line is halved, while it remains the same side from the second straight line. [hide=original wording] На плоскости даны две прямые, пересекающиеся под острым углом. В направлении одной из прямых производится сжатие 1 с коэффициентом 1/2. Доказать, что найдется точка, расстояние от которой до точки пересечения прямых увеличится. Здесь имеется в виду преобразование, при котором каждая точка перемещается параллельно одной прямой так, что её расстояние до второй прямой уменьшается вдвое, причём она остаётся по ту же самую сторону от второй прямой[/hide]

Three distinct points $A, B$, and $C$ lie on a circle with centre at $O$. The triangles $AOB, BOC$ , and $COA$ have equal area. What are the possible measures of the angles of the triangle $ABC$ ?

Let $ABC$ be a triangle with $AB<AC$ and let $M$ be the midpoint of $AC$. A point $P$ (other than $B$) is chosen on the segment $BC$ in such a way that $AB=AP$. Let $D$ be the intersection of $AC$ with the circumcircle of $\bigtriangleup ABP$ distinct from $A$, and $E$ be the intersection of $PM$ with the circumcircle of $\bigtriangleup ABP$ distinct from $P$. Let $K$ be the intersection of lines $AP$ and $DE$. Let $F$ be a point on $BC$ (other than $P$) such that $KP=KF$. Show that $C,\ D,\ E$ and $F$ lie on the same circle.

The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

Let $ ABC$ be a triangle with $ \angle A < 60^\circ$. Let $ X$ and $ Y$ be the points on the sides $ AB$ and $ AC$, respectively, such that $ CA \plus{} AX \equal{} CB \plus{} BX$ and $ BA \plus{} AY \equal{} BC \plus{} CY$ . Let $ P$ be the point in the plane such that the lines $ PX$ and $ PY$ are perpendicular to $ AB$ and $ AC$, respectively. Prove that $ \angle BPC < 120^\circ$.

A parabola is inscribed in equilateral triangle $ABC$ of side length $1$ in the sense that $AC$ and $BC$ are tangent to the parabola at $A$ and $B$, respectively. Find the area between $AB$ and the parabola.

In the tetrahedron $ABCD,$ on the continuation of the edges $AB, AC$ and $AD$, three points were marked for point $A{},$ located from $A{}$ at a distance equal to the semi-perimeter of the triangle $BCD.$ Similarly, we marked three points corresponding to vertices $B, C$ and $D.$ Prove that if there is a sphere touching all the edges of the tetrahedron $ABCD$, then the marked 12 points lie on the same sphere. [i]Proposed by V. Alexandrov[/i]

A convex quadrilateral $ABCD$ with $AC \neq BD$ is inscribed in a circle with center $O$. Let $E$ be the intersection of diagonals $AC$ and $BD$. If $P$ is a point inside $ABCD$ such that $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$, prove that $O$, $P$ and $E$ are collinear.

Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$. Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.

Let $\overline{AB}$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $\overline{AB}$. Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$. Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$. Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. Line $PQ$ intersects $\omega$ at $X$ and $Y$. Assume that $AP=5$, $PB=3$, $XY=11$, and $PQ^2 = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.