Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.). The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.

Let $ABCD$ be a convex quadrilateral. The bisector of the angle $ACD$ intersects $BD$ at point $E$. It is known that $\angle CAD = \angle BCE= 90^o$. Prove that the $AC$ is the bisector of the angle $BAE$ . (Nikolay Nikolay)

The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one? [img]https://1.bp.blogspot.com/-aeqefhbBh5E/XzcBjhgg7sI/AAAAAAAAMXM/B0qSgWDBuqc3ysd-mOitP1LarOtBdJJ3gCLcBGAsYHQ/s0/2004%2BMohr%2Bp1.png[/img]

Vertex $ E$ of equilateral $ \triangle{ABE}$ is in the interior of unit square $ ABCD$. Let $ R$ be the region consisting of all points inside $ ABCD$ and outside $ \triangle{ABE}$ whose distance from $ \overline{AD}$ is between $ \frac{1}{3}$ and $ \frac{2}{3}$. What is the area of $ R$? $ \textbf{(A)}\ \frac{12\minus{}5\sqrt3}{72} \qquad \textbf{(B)}\ \frac{12\minus{}5\sqrt3}{36} \qquad \textbf{(C)}\ \frac{\sqrt3}{18} \qquad \textbf{(D)}\ \frac{3\minus{}\sqrt3}{9} \qquad \textbf{(E)}\ \frac{\sqrt3}{12}$

Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus $ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.

A regular octagon $ ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ ABEF$? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)+fontsize(6pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);[/asy]$ \textbf{(A)}\ 1\minus{}\frac{\sqrt2}{2} \qquad \textbf{(B)}\ \frac{\sqrt2}{4} \qquad \textbf{(C)}\ \sqrt2\minus{}1 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac{1\plus{}\sqrt2}{4}$

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

A circle centered at $ A$ with a radius of 1 and a circle centered at $ B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is [asy] size(220); real r1 = 1; real r2 = 3; real r = (r1*r2)/((sqrt(r1)+sqrt(r2))**2); pair A=(0,r1), B=(2*sqrt(r1*r2),r2); dot(A); dot(B); draw( circle(A,r1) ); draw( circle(B,r2) ); draw( (-1.5,0)--(7.5,0) ); draw( A -- (A+dir(210)*r1) ); label("$1$", A -- (A+dir(210)*r1), N ); draw( B -- (B+r2*dir(330)) ); label("$4$", B -- (B+r2*dir(330)), N ); label("$A$",A,dir(330)); label("$B$",B, dir(140)); draw( circle( (2*sqrt(r1*r),r), r )); [/asy] $ \displaystyle \textbf{(A)} \ \frac {1}{3} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {5}{12} \qquad \textbf{(D)} \ \frac {4}{9} \qquad \textbf{(E)} \ \frac {1}{2}$

An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.

Let $ACD$ and $BCD$ be acute-angled triangles located in different planes. Let $G$ and $H$ be the centroid and the orthocenter respectively of the $BCD$ triangle; Similarly let $G'$ and $H'$ be the centroid and the orthocenter of the $ACD$ triangle. Knowing that $HH'$ is perpendicular to the plane $(ACD)$, show that $GG' $ is perpendicular to the plane $(BCD)$.

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. Ray $O_1B$ meets $S_2$ again at $F$, and ray $O_2B$ meets $ S_1$ again at $E$. The line through $B$ parallel to $ EF$ intersects $S_1$ and $S_2$ again at $M$ and $N$, respectively. Prove that $MN = AE +AF$.

$\triangle ABC$ is an equilateral triangle. $D$ is a point on $\overline{AC}$, and $E$ is a point on $\overline{BD}$. Let $P$ and $Q$ be the circumcenters of $\triangle ABD$ and $\triangle AED$, respectively. Prove that $ \triangle EPQ$ is an equilateral triangle if and only if $ \overline{AB} \perp \overline{CE}$.

Let $D$ be a point on the side $BC$ of an equilateral triangle $ABC$ where $D$ is different than the vertices. Let $I$ be the excenter of the triangle $ABD$ opposite to the side $AB$ and $J$ be the excenter of the triangle $ACD$ opposite to the side $AC$. Let $E$ be the second intersection point of the circumcircles of triangles $AIB$ and $AJC$. Prove that $A$ is the incenter of the triangle $IEJ$.

On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.

Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n\geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum\limits_{n=1}^{\infty}C_{n-1}C_n = 6p$. Find $p$.

Points $M$ and $N$ are given on sides $AD$ and $BC$ of rhombus $ABCD$, respectively. Line $MC$ intersects line $BD$ in point $T$, line $MN$ intersects line $BD$ in point $U$, line $CU$ intersects line $AB$ in point $Q$ and line $QT$ intersects line $CD$ in $P$. Prove that triangles $QCP$ and $MCN$ have equal area

Determine positive integers $p, q$, and $r$ such that the diagonal of a block consisting of $p\times q\times r$ unit cubes passes through exactly $1984$ of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)

In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$.

A point $O$ lies inside of the square $ABCD$. Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\circ}$ does not exceed $45^{\circ}$.

Three concentric circles with common center $O$ are cut by a common chord in successive points $A, B, C$. Tangents drawn to the circles at the points $A, B, C$ enclose a triangular region. If the distance from point $O$ to the common chord is equal to $p$, prove that the area of the region enclosed by the tangents is equal to \[\frac{AB \cdot BC \cdot CA}{2p}\]

Let $ A^{\prime}$ be an arbitrary point on the side $ BC$ of a triangle $ ABC$. Denote by $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ the circles simultanously tangent to $ AA^{\prime}$, $ A^{\prime}B$, $ \Gamma$ and $ AA^{\prime}$, $ A^{\prime}C$, $ \Gamma$, respectively, where $ \Gamma$ is the circumcircle of $ ABC$. Prove that $ \mathcal{T}_{A}^{b}$, $ \mathcal{T}_{A}^{c}$ are congruent if and only if $ AA^{\prime}$ passes through the Nagel point of triangle $ ABC$. ([i]If $ M,N,P$ are the points of tangency of the excircles of the triangle $ ABC$ with the sides of the triangle $ BC$, $ CA$ and $ AB$ respectively, then the Nagel point of the triangle is the intersection point of the lines $ AM$, $ BN$ and $ CP$[/i].)

a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$. Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$. b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$. Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.

Suppose the transformation $T$ acts on points in the plane like this: $$T(x, y) = \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2 + y^2}\right).$$ Determine the area enclosed by the set of points of the form $T(x, y)$, where $(x, y)$ is a point on the edge of a length-$2$ square centered at the origin with sides parallel to the axes.

The sum of the lengths of the twelve edges of a rectangular box is $140$, and the distance from one corner of the box to the farthest corner is $21$. The total surface area of the box is $\text{(A)}\ 776 \qquad \text{(B)}\ 784 \qquad \text{(C)}\ 798 \qquad \text{(D)}\ 800 \qquad \text{(E)}\ 812$

Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$, $AO = CO$. Points $P$ and $Q$ are marked on the segments $AO$ and $CO$, respectively, such that $PO = OQ$. Let $N$ and $K$ be the intersection points of the sides $AB$, $CD$, and the lines $DP$ and $BQ$ respectively. Prove that the points $N$, $O$, and $K$ are colinear.