Let be given two concentric circles of radii $R$ and $R_1 > R$. Let quadrilateral $ABCD$ is inscribed in the smaller circle and let the rays $CD, DA, AB, BC$ meet the larger circle at $A_1, B_1, C_1, D_1$ respectively. Prove that $$ \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}$$ where $\sigma(P)$ denotes the area of a polygon $P.$

A (convex) trapezoid $ABCD$ is good, if it is inscribed in a circle, sides $AB$ and $CD$ are the bases and $CD$ is shorter than $AB$. For a good trapezoid $ABCD$ the following terms are defined: $\bullet$ The parallel to $AD$ passing through $B$ intersects the extension of side $CD$ at point $S$. $\bullet$ The two tangents passing through $S$ on the circumircle of the trapezoid touch the circle at $E$ and $F$, where $E$ lies on the same side of the straight line $CD$ as $A$. Give the simplest possible equivalent condition (expressed in side lengths and / or angles of the trapezoid) so that with a good trapezoid $ABCD$ the two angles $\angle BSE$ and $\angle FSC$ have the same measure. (Walther Janous)

Let $ABC$ be a triangle with medians $m_a$ , $m_b$, $m_c$. Prove that: (a) There is a triangle with side lengths $m_a$ ,$m_b$, $m_c$. (b) This triangle is similar to $ABC$ if and only if the squares of the side lengths of triangle $ABC$ form an arithmetical sequence.

There are $n$ people standing on a circular track. We want to perform a number of [i]moves[/i] so that we end up with a situation where the distance between every two neighbours is the same. The [i]move[/i] that is allowed consists in selecting two people and asking one of them to walk a distance $d$ on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity $d$ can vary from move to move. Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most $n-1$ moves.

$AB$ and $AC$ are tangents to a circle $\omega$ with center $O$ at $B,C$ respectively. Point $P$ is a variable point on minor arc $BC$. The tangent at $P$ to $\omega$ meets $AB,AC$ at $D,E$ respectively. $AO$ meets $BP,CP$ at $U,V$ respectively. The line through $P$ perpendicular to $AB$ intersects $DV$ at $M$, and the line through $P$ perpendicular to $AC$ intersects $EU$ at $N$. Prove that as $P$ varies, $MN$ passes through a fixed point.

A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing? [asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1)--(6,5)--(4,4)); draw((6,5)--(2,5)--(0,4)); draw(ellipse((5,2.5),0.5,1)); fill(ellipse((3,4.5),1,0.25),black); fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black); fill(ellipse((3,5.25),1,0.25),black); [/asy] $\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and find the ratio of their areas.

Consider the intersection of an ellipsoid with a plane $ \sigma$ passing through its center $ O$. On the line through the point $ O$ perpendicular to $ \sigma$, mark the two points at a distance from $ O$ equal to the area of the intersection. Determine the loci of the marked points as $ \sigma$ runs through all such planes. [i]L. Tamassy[/i]

A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.

Let $ABC$ be a triangle inscribed in circle $(O)$ such that points $B, C$ are fixed, while $A$ moves on major arc $BC$ of $(O)$. The tangents through $B$ and $C$ to $(O)$ intersect at $P$. The circle with diameter $OP$ intersects $AC$ and $AB$ at $D$ and $E$, respectively. Prove that $DE$ is tangent to a fixed circle whose radius is half the radius of $(O)$.

Let \(ABCDE\) be a regular pentagon. Let \(P\) be a variable point on the interior of segment \(AB\) such that \(PA\ne PB\). The circumcircles of \(\triangle PAE\) and \(\triangle PBC\) meet again at \(Q\). Let \(R\) be the circumcenter of \(\triangle DPQ\). Show that as \(P\) varies, \(R\) lies on a fixed line. [i]Proposed by Karthik Vedula[/i]

Let the incircle of the triangle $ABC$ touch $[BC]$ at $D$ and $I$ be the incenter of the triangle. Let $T$ be midpoint of $[ID]$. Let the perpendicular from $I$ to $AD$ meet $AB$ and $AC$ at $K$ and $L$, respectively. Let the perpendicular from $T$ to $AD$ meet $AB$ and $AC$ at $M$ and $N$, respectively. Show that $|KM|\cdot |LN|=|BM|\cdot|CN|$.

The point $M$ varies on the segment $AB$ that measures $2$ m. a) Find the equation and the graphical representation of the locus of the points of the plane whose coordinates, $x$, and $y$, are, respectively, the areas of the squares of sides $AM$ and $MB$ . b) Find out what kind of curve it is. (Suggestion: make a $45^o$ axis rotation). c) Find the area of the enclosure between the curve obtained and the coordinate axes.

Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$. We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$. Compute $a + b$.

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

Let $ABC$ be an acute triangle with $| AB | > | AC |$. Let $D$ be the foot of the altitude from $A$ to $BC$, let $K$ be the intersection of $AD$ with the internal bisector of angle $B$, Let $M$ be the foot of the perpendicular from $B$ to $CK$ (it could be in the extension of segment $CK$) and$ N$ the intersection of $BM$ and $AK$ (it could be in the extension of the segments). Let $T$ be the intersection of$ AC$ with the line that passes through $N$ and parallel to $DM$. Prove that $BM$ is the internal bisector of the angle $\angle TBC$

A quadrilateral $ABCD$ is circumscribed about a circle. Lines $AC$ and $DC$ meet at point $E$ and lines $DA$ and $BC$ meet at $F$, where $B$ is between $A$ and $E$ and between $C$ and $F$. Let $I_1, I_2$ and $I_3$ be the incenters of triangles $AFB, BEC$ and $ABC$, respectively. The line $I_1I_3$ intersects $EA$ at $K$ and $ED$ at $L$, whereas the line $I_2I_3$ intersects $FC$ at $M$ and $FD$ at $N$. Prove that $EK = EL$ if and only if $FM = FN$

A square $ABCD,AB=s=1$ is given in the plane with its center $S$. Furthermore, points $E,F$ are given on the rays opposite to $CB,DA$, respectively, $CE=a,DF=b$. Determine all triangles $XYZ$ such that $X,Y,Z$ lie in this order on segments $CD,AD,BC$ and $E,S,F$ lie on lines $XY,YZ,ZX$ respectively. Discuss conditions of solvability in terms of $a,b,s$ and unknown $x=CX$.

In rectangle $ ABCD$, we have $ AB\equal{}8$, $ BC\equal{}9$, $ H$ is on $ \overline{BC}$ with $ BH\equal{}6$, $ E$ is on $ \overline{AD}$ with $ DE\equal{}4$, line $ EC$ intersects line $ AH$ at $ G$, and $ F$ is on line $ AD$ with $ \overline{GF}\perp\overline{AF}$. Find the length $ GF$. [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0); draw(D--A--B--C--D--F--G--Ep); draw(A--G); label("$F$",F,W); label("$G$",G,W); label("$C$",C,WSW); label("$H$",H,NNE); label("$6$",(6,8),N); label("$B$",B,NE); label("$A$",A,SW); label("$E$",Ep,S); label("$4$",(2,0),S); label("$D$",D,S);[/asy]$ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

10.7 A quadrilateral $ABCD$ is circumscribed around the circle $\omega$ centered at $I$ and inscribed into the circle $\Gamma$. The lines $AB, CD$ meet at point $P$, and the lines $BC, AD$ meet at point $Q$. Prove that the circles $\odot(PIQ)$ and $\Gamma$ are orthogonal.

A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$. [i]Proposed by S. Berlov[/i]

Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$, $\omega_{B}$ to $BC$ and $BA$, $\omega_{C}$ to $CA$ and $CB$, and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$. If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is $5$ inches from each wall which that ball touches and $10$ inches from the floor, then the sum of the diameters of the balls is $\textbf{(A) }20\text{ inches}\qquad\textbf{(B) }30\text{ inches}\qquad\textbf{(C) }40\text{ inches}\qquad$ $\textbf{(D) }60\text{ inches}\qquad \textbf{(E) }\text{not determined by the given information}$

Assume that the bisecting plane of the dihedral angle at edge $AB$ of the tetrahedron $ABCD$ meets the edge $CD$ at point $E$. Denote by $S_1, S_2, S_3$, respectively the areas of the triangles $ABC, ABE$, and $ABD$. Prove that no tetrahedron exists for which $S_1, S_2, S_3$ (in this order) form an arithmetic or geometric progression.

Let $ABCD$ be a parallelogram and point $M$ be the midpoint of $[AB]$ so that the quadrilateral $MBCD$ is cyclic. If $N$ is the point of intersection of the lines $DM$ and $BC$, and $P \in BC$, then prove that the ray $(DP$ is the angle bisector of $\angle ADM$ if and only if $PC = 4BC$.