Let the triangle $ ADB_1$ s.t. $ m(\angle DAB_1)\ne 90^\circ$.On the sides of this triangle externally are constructed the squares $ ABCD$ and $ AB_1C_1D_1$ with centers $ O_1$ and $ O_2$, respectively.Prove that the circumcircles of the triangles $ BAB_1$, $ DAD_1$ and $ O_1AO_2$ share a common point, that differs from $ A$.

Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$ and $AB$ its shortest side. Denote by $H$ the intersection of the altitudes of triangle $ABC$. Let $K$ be the circle through $A$ with centre $B$. Let $D$ be the other intersection of $K$ and $AC$. Let $K$ intersect the circumcircle of $BCD$ again at $E$. If $F$ is the intersection of $DE$ and $BH$, show that $BD$ is tangent to the circle through $D$, $F$, and $H$.

On the side $AC$ of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.

There are pieces in the shape of an equilateral triangle with sides $1, 2, 3, 4, 5$ and $6$ ($50$ pieces of each size). You want to build an equilateral triangle of side $7$ using some of these pieces, without gaps or overlaps. What is the least number of pieces needed?

Let $\omega_1$ and $\omega_2$ be circles that intersect at points $A$ and $B$ and let $O_1$ and $O_2$ be their respective centers. We take $M$ in $\omega_1$ and $N$ in $\omega_2$ on the same side as $B$ with respect to segment $O_1O_2$, such that $MO_1\parallel BO_2$ and $BO_1 \parallel NO_2$. Draw the tangents to $\omega_1$ and $\omega_2$ through $M$ and $N$ respectively, which intersect at $K$. Show that $A$, $B$ and $K$ are collinear.

On a circle with diameter $AB$ we marked an arbitrary point $C$, which does not coincide with $A$ and $B$. The tangent to the circle at point $A$ intersects the line $BC$ at point $D$. Prove that the tangent to the circle at point $C$ bisects the segment $AD$.

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$. Prove that the lines $AP$ and $CS$ are parallel.

Given 3 non-collinear points $A,B,C$. For each point $M$ in the plane ($ABC$) let $M_1$ be the point symmetric to $M$ with respect to $AB$, $M_2$ be the point symmetric to $M_1$ with respect to $BC$ and $M'$ be the point symmetric to $M_2$ with respect to $AC$. Find all points $M$ such that $MM'$ obtains its minimum. Let this minimum value be $d$. Prove that $d$ does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is $BC$, $CA$, $AB$. In the first part the order of axes we chose $AB$, $BC$, $CA$, and the second part of the problem states that the value $d$ doesn't depend on this order).

A square of side $1$ is covered with $m^2$ rectangles. Show that there is a rectangle with perimeter at least $\frac{4}{m}$.

$\Delta ABC$ is isosceles with a circumscribed circle $\omega (O)$. Let $H$ be the foot of the altitude from $C$ to $AB$ and let $M$ be the middle point of $AB$. We define a point $X$ as the second intersection point of the circle with diameter $CM$ and $\omega$ and let $XH$ intersect $\omega$ for a second time in $Y$. If $CO\cap AB=D$, then prove that the circumscribed circle of $\Delta YHD$ is tangent to $\omega$.

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units? [asy] size(4cm); filldraw(scale(2)*unitcircle,gray,black); filldraw(shift(-1,0)*unitcircle,white,black); filldraw(shift(1,0)*unitcircle,white,black); [/asy] $\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}$

Two quarter-circles touch as shown. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png[/img]

We consider a right trapezoid $ABCD$, in which $AB||CD,AB>CD,AD\perp AB$ and $AD>CD$. The diagonals $AC$ and $BD$ intersect at $O$. The parallel through $O$ to $AB$ intersects $AD$ in $E$ and $BE$ intersects $CD$ in $F$. Prove that $CE\perp AF$ if and only if $AB\cdot CD=AD^2-CD^2$ .

There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?

Let $n \geq 6$ and $3 \leq p < n - p$ be two integers. The vertices of a regular $n$-gon are colored so that $p$ vertices are red and the others are black. Prove that there exist two congruent polygons with at least $[p/2] + 1$ vertices, one with all the vertices red and the other with all the vertices black.

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]

Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$. Point $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$. Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$. Prove that $PQRS$ is a cyclic quadrilateral. [i]Proposed by Evan O'Dorney[/i]

In the figure below, the centers of the six congruent circles form a regular hexagon with side length 2. [asy] import graph; pair A = 2dir(0); pair B = 2dir(60); pair C = 2dir(120); pair D = 2dir(180); pair E = 2dir(240); pair F = 2dir(300); path hexagon = A -- B -- C -- D -- E -- F -- cycle; fill(hexagon, gray); filldraw(Circle(A, 1), white); filldraw(Circle(B, 1), white); filldraw(Circle(C, 1), white); filldraw(Circle(D, 1), white); filldraw(Circle(E, 1), white); filldraw(Circle(F, 1), white); draw(hexagon); [/asy] Adjacent circles are tangent to each other. What is the area of the shaded region?

How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?

assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.

In convex quadrilateral $ABCD$, let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$, respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$, and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$. Prove that $A$, $B$, $R$, and $S$ are concyclic. [i]Tiger Zhang[/i]

In the triangle $ABC$ on the side $AC$ the points $F$ and $L$ are selected so that $AF = LC <\frac{1}{2} AC$. Find the angle $ \angle FBL $ if $A {{B} ^ {2}} + B {{C} ^ {2}} = A {{L} ^ {2}} + L {{C } ^ {2}}$ (Zhidkov Sergey)

Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.

Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and[list][*]if the card is black, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance 1 away from the vertex;[*]if the card is white, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance $\sqrt2$ away from the vertex.[/list]When Tessa finishes drawing all cards of the deck, what is the maximum possible value of a number that is on the cube?