The plane has a circle $\omega$ and a point $A$ outside it. For any point $C$, the point $B$ on the circle $\omega$ is defined such that $ABC$ is an equilateral triangle with vertices $A, B$ and $C$ listed clockwise. Prove that if point $B$ moves along the circle $\omega$, then point $C$ also moves along a circle.

Extensions of opposite sides of a convex quadrilateral $ABCD$ intersect at points $P$ and $Q$. Points are marked on the sides of $ABCD$ (one per side), which are the vertices of a parallelogram with a side parallel to $PQ$. Prove that the intersection point of the diagonals of this parallelogram lies on one of the diagonals of quadrilateral $ABCD$. (E. Bakaev)

The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are solid. The security guard guarding the museum has two favourite spots (points $A$ and $B$) because one can see the whole area of the museum standing at either point. Is it true that from any point of the $AB$ section one can see the whole museum?

In $\triangle ABC$ with integer side lengths, \[ \cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}. \] What is the least possible perimeter for $\triangle ABC$? $\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 44$

An $11\times 11$ chess board is covered with one $\boxed{ }$ shaped and forty $\boxed{ }\boxed{ }\boxed{ }$ shaped tiles. Determine the squares where $\boxed{}$ shaped tile can be placed.

The points $X, \, \, Y$are selected on the sides $AB$ and $AD$ of the convex quadrilateral $ABCD$, respectively. Find the ratio $AX \, \,: \, \, BX$ if you know that $CX || DA$, $DX || CB$, $BY || CD$ and $CY || BA$.

Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.

In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.

Prove that the side $AB$ of a convex quadrilateral $ABCD$ is less than its diagonal $AC$ if $|AB|+|BC| \le |AC| +| CD|$.

Reconstruct a triangle, given the incenter, the midpoint of some side and the foot of the altitude drawn on this side.

A ball moves endlessly on a circular billiard table. When it hits the edge it is reflected. Show that if it passes through a point on the table three times, then it passes through it infinitely many times.

The three sides and the area of a triangle are integers. What is the smallest value of the area of this triangle?

The incircle of a right triangle $ABC$ touches its hypotenuse $BC$ at point $D$. The line $AD$ intersects the circumscribed circle at point $X$. Prove that $ |BX-CX| \geqslant |AD - DX|$.

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

Consider a convex quadrilateral, and the incircles of the triangles determined by one of its diagonals. Prove that the tangency points of the incircles with the diagonal are symmetrical with respect to the midpoint of the diagonal if and only if the line of the incenters passes through the crossing point of the diagonals. [i]Dan Schwarz[/i]

If a square is drawn externally on each side of a parallelogram, prove that: (a) The quadrilateral formed by the centers of these squares is also a square. (b) The diagonals of the new square formed are concurrent with the diagonals of the original parallelogram.

In a right triangle with the length legs $b$ and $c$, and the length hypotenuse $a$, the ratio between the length of the hypotenuse and the length of the diameter of the inscribed circle does not exceed $1 + \sqrt2$. Determine the numerical value of the expression of $E =\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}$.

Given a convex $n$-gon with pairwise (mutually) non-parallel sides and a point inside it. Prove that there are not more than $n$ straight lines coming through that point and halving the area of the $n$-gon.

[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

A quadrilateral is circumscribed around a circle. Its diagonals intersect at the center of the circle. Prove that the quadrilateral is a rhombus.

Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6, DB = 2$, and the ratio $\tfrac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5.$ Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

A square with side $1$ contains a non-self-intersecting polyline of length at least $200$. Prove that there is a straight line parallel to the side of the square that has at least $101$ points in common with this polyline.

A wheel with a rubber tire has an outside diameter of $ 25$ in. When the radius has been decreased a quarter of an inch, the number of revolutions in one mile will: $ \textbf{(A)}\ \text{be increased about }2\% \qquad\textbf{(B)}\ \text{be increased about }1\%$ $ \textbf{(C)}\ \text{be increased about }20\% \qquad\textbf{(D)}\ \text{be increased about }\frac {1}{2}\% \qquad\textbf{(E)}\ \text{remain the same}$

Let $ABC$ be a triangle. Let $B'$ be the symmetric of $B$ with respect to $C, C'$ the symmetry of $C$ with respect to $A$ and $A'$ the symmetry of $A$ with respect to $B$. a) Prove that the area of triangle $AC'A'$ is twice the area of triangle $ABC$. b) If we delete points $A, B, C$, how can they be reconstituted? Justify your reasoning.