In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD, O, G, H, J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF$. Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ become arbitrarily close to: [asy] size((270)); draw((0,0)--(10,0)..(5,5)..(0,0)); draw((5,0)--(5,5)); draw((9,3)--(1,3)--(1,1)--(9,1)--cycle); draw((9.9,1)--(.1,1)); label("O", (5,0), S); label("a", (7.5,0), S); label("G", (5,1), SE); label("J", (5,5), N); label("H", (5,3), NE); label("E", (1,3), NW); label("L", (1,1), S); label("C", (.1,1), W); label("F", (9,3), NE); label("M", (9,1), S); label("D", (9.9,1), E); [/asy] $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{1}{\sqrt{2}}+\frac{1}{2} \qquad\textbf{(E)}\ \frac{1}{\sqrt{2}}+1$

Four points $ A, B, C, D$ are chosen on a circle in such a way that arcs $ AB, BC,$ and $ CD$ are of the same length and the $ arc DA$ is longer than these three. Line $ AD$ and the line tangent to the circle at $ B$ intersect at $ E$. Let $ F$ be the other endpoint of the diameter starting at $ C$ of the circle. Prove that triangle $ DEF$ is equilateral.

In the triangle $ABC, AC=BC, \angle C=90^o, D$ is the midpoint of $BC, E$ is the point on $AB$ such that $AD$ is perpendicular to $CE$. Prove that $AE=2EB$.

Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. [i]M. Kungodjin[/i]

The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$. The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$.The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$. Prove that $CF$ is the altitude of triangle $ABC$.

Find the maximal positive number $r$ with the following property: If all altitudes of a tetrahedron are $\geq 1$, then a sphere of radius $r$ fits into the tetrahedron.

On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$. The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$. Prove that: (a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$; (b) $P,A,Q$ are collinear. [i]Virginia Tica, Vasile Tica[/i]

Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.

Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.

Prove that if a sphere can be inscribed in a convex polyhedron and each face of this polyhedron can be painted in one of two colors such that any two faces sharing a common edge are of different colors, then the sum of the areas of the faces of one color is equal to the sum of the areas of the faces of the other color.

5. In tetrahedron $ABCD$ following equalities hold: $\angle BAC+\angle BDC=\angle ABD+\angle ACD$ $\angle BAD+\angle BCD=\angle ABC+\angle ADC$ Prove that center of sphere circumscribed about ABCD lies on a line through midpoints of $AB$ and $CD$.

Let $ABCD$ be an isosceles trapezoid with $AB=AD=BC, AB//CD, AB>CD$. Let $E= AC \cap BD$ and $N$ symmetric to $B$ wrt $AC$. Prove that the quadrilateral $ANDE$ is cyclic.

Is it possible to cut a regular triangular prism into two equal pyramids?

Determine all integers $ n > 3$ for which there exist $ n$ points $ A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $ r_{1},\cdots ,r_{n}$ such that for $ 1\leq i < j < k\leq n$, the area of $ \triangle A_{i}A_{j}A_{k}$ is $ r_{i} \plus{} r_{j} \plus{} r_{k}$.

An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle. by Yu.Blinkov

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$

$ABCDE$ is a regular pentagon (with vertices in that order) inscribed in a circle of radius $1$. Find $AB \cdot AC$.

Alex marked one point on each of the six interior faces of a hollow unit cube. Then he connected by strings any two marked points on adjacent faces. Prove that the total length of these strings is at least $6\sqrt2$.

Let $O$ be a point inside the parallelogram $ABCD$ such that $\angle AOB + \angle COD = \angle BOC + \angle AOD$. Prove that there exists a circle $k$ tangent to the circumscribed circles of the triangles $\vartriangle AOB, \vartriangle BOC, \vartriangle COD$ and $\vartriangle DOA$.

Let $AD$ be the altitude, $AE$ be the median, and $O$ be the circumcenter of a triangle $ABC.$ Points $X$ and $Y$ are selected inside the triangle such that $\angle BAX=\angle CAY,$ $OX\perp AX,$ and $OY\perp AY.$ Prove that points $D,E,X,Y$ are concyclic. (M. Kurskiy)

Let $AH_A, BH_B, CH_C$ be the altitudes of the triangle $ABC$. Points $A_1$ and $C_1$ are the projections of the point $H_B$ onto the sides $AB$ and $BC$, respectively. $B_1$ is the projection of $B$ onto $H_AH_C$. Prove that the diameter of the circumscribed circle of $\triangle A_1B_1C_1$ is equal to $BH_B$. [i]Proposed by Anton Trygub[/i]

In the triangle $ABC$, $X$ lies on the side $BC$, $Y$ on the side $CA$, and $Z$ on the side $AB$ with $YX \| AB, ZY \| BC$, and $XZ \| CA$. Show that $X,Y$, and $Z$ are the midpoints of the respective sides of $ABC$.

$6$ points are selected on the plane, none of which $3$ lie on one straight line, and all pairwise segments connecting these points are plotted. Some of the sections are plotted in red and others in blue. Prove that any three of the given points are the vertices of a triangle with sides of the same color.

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A'$, two circles through $B$ at $B'$ , two circles at $C$ at $C'$ and the two circles at $D$ at $D'$. Suppose the points $A',B',C'$ and $D'$ are distinct. Prove quadrilateral $A'B'C'D'$ is similar to $ABCD$.