Let side $BC$ of $\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively. $DG$ and $EF$ intersect at $M$. Prove that $AM \perp BC$.

Given the nine-sided regular polygon $ A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $ \{A_1,A_2,...A_9\}$? $ \textbf{(A)} \ 30 \qquad \textbf{(B)} \ 36 \qquad \textbf{(C)} \ 63 \qquad \textbf{(D)} \ 66 \qquad \textbf{(E)} \ 72$

Let $ABC$ be a triangle inscribed in the circle $(O)$ and $P$ is a point inside the triangle $ABC$. Let $D$ be a point on $(O)$ such that $AD \perp AP$. The line $CD$ cuts the perpendicular bisector of $BC$ at $M$. The line $AD$ cuts the line passing through $B$ and is perpendicular to $BP$ at $Q$. Let $N$ be the reflection of $Q$ through $M$. Prove that $CN \perp CP$.

The diagram below shows a large square with each of its sides divided into four equal segments. The shaded square whose sides are diagonals drawn to these division points has area $13$. Find the area of the large square. [img]https://cdn.artofproblemsolving.com/attachments/8/3/bee223ef39dea493d967e7ebd5575816954031.png[/img]

For which $n\ge 2$ is it possible to find $n$ pairwise non-similar triangles $A_1, A_2,\ldots , A_n$ such that each of them can be divided into $n$ pairwise non-similar triangles, each of them similar to one of $A_1,A_2 ,\ldots ,A_n$?

Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.

Let $ k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $ \frac {AL}{AD} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k$. Denote $ (O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C. 1) Prove that when $ k \equal{} \frac{1}{2}$, three circles $ (O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles. 2) Find all values of k such that three circles $ (O_1),(O_2),(O_3)$ have exactly two common points

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$

It is known that a square can be inscribed in a given right trapezoid so that each of its vertices lies on the corresponding side of the trapezoid (none of the vertices of the square coincides with the vertex of the trapezoid). Construct this inscribed square with a compass and a ruler. (Maria Rozhkova)

$B$ is the center of unit circle. $A,C$ are points on the circle (the order of $A,B,C$ is clockwise), and $\angle ABC=2\alpha(0<\alpha<\frac{\pi}{3})$. Then we will rotate $\triangle ABC$ anticlockwise. In the first rotation, $A$ is the center of rotation, the result is that $B$ is on the circle. In the second rotation, $B$ is the center of rotation, the result is that $C$ is on the circle. In the third rotation, $C$ is the center of rotation, the result is that $A$ is on the circle. ... After we rotate for $100$ times, the distance $A$ travelled is $\text{(A)}22\pi(1+\sin\alpha)-66\alpha\qquad\text{(B)}\frac{67}{3}\pi\qquad\text{(C)}22\pi+\frac{68}{3}\pi\sin\alpha-66\alpha\qquad\text{(D)}33\pi-66\alpha$

There is a rhombus with acute angle $b$ and side $a$. Two parallel lines, the distance between which is equal to the height of the rhombus, intersect all four sides of the rhombus. What can be the sum of the perimeters of two triangles cut off from a rhombus by straight lines? (These two triangles lie outside the strip between parallel lines.)

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and incenter $I$. The midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$ is denoted $S$. Points $E, F$ were chosen on line $OI$ for which $BE$ and $CF$ are both perpendicular to $OI$. Point $X$ was chosen so that $XE\perp AC$ and $XF\perp AB$. Point $Y$ was chosen so that $YE\perp SC$ and $YF\perp SB$. $D$ was chosen on $BC$ so that $DI\perp BC$. Prove that $X$, $Y$, and $D$ are collinear.

Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

Let $ \vartriangle ABC $ be a triangle with $ \angle ACB = 60º $. Let $ E $ be a point inside $ \overline {AC} $ such that $ CE <BC $. Let $ D $ over $ \overline {BC} $ such that $$ \frac {AE} {BD} = \frac {BC} {CE} -1 .$$ Let us call $ P $ the intersection of $ \overline {AD} $ with $ \overline {BE} $ and $ Q $ the other point of intersection of the circumcircles of the triangles $ AEP $ and $ BDP $. Prove that $QE \parallel BC $.

Let $\vec{v_1},\vec{v_2},\vec{v_3},\vec{v_4}$ and $\vec{v_5}$ be vectors in three dimensions. Show that for some $i,j$ in $1,2,3,4,5$, $\vec{v_i}\cdot \vec{v_j}\ge 0$.

Let $(O)$ denote a circle with a chord $AB$, and let $W$ be the midpoint of the minor arc $AB$. Let $C$ stand for an arbitrary point on the major arc $AB$. The tangent to the circle $(O)$ at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$, respectively. The lines $W X$ and $W Y$ meet $AB$ at points $N$ and $M$ , respectively. Does the length of segment $NM$ depend on position of $C$ ?

Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$. [hide="comment"] [i]Edited by Orl.[/i] [/hide]

Let $x=\cos 36^{\circ} - \cos 72^{\circ}$. Then $x$ equals $ \textbf{(A)}\ \frac{1}{3} \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ 3-\sqrt{6} \qquad\textbf{(D)}\ 2\sqrt{3}-3 \qquad\textbf{(E)}\ \text{none of these} $

In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.

In a triangle $ABC$, the bisector of angle $BAC$ intersects $BC$ at $D$. The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$. Line $BM$ meets $\Gamma$ again at $P$. Prove that line $AP$ is a median of $\triangle ABD.$

The area of the smallest square that will contain a circle of radius 4 is $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 128$

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

Let $ ABC$ be a triangle. The $ A$-excircle of triangle $ ABC$ has center $ O_{a}$ and touches the side $ BC$ at the point $ A_{a}$. The $ B$-excircle of triangle $ ABC$ touches its sidelines $ AB$ and $ BC$ at the points $ C_{b}$ and $ A_{b}$. The $ C$-excircle of triangle $ ABC$ touches its sidelines $ BC$ and $ CA$ at the points $ A_{c}$ and $ B_{c}$. The lines $ C_{b}A_{b}$ and $ A_{c}B_{c}$ intersect each other at some point $ X$. Prove that the quadrilateral $ AO_{a}A_{a}X$ is a parallelogram. [i]Remark.[/i] The $ A$[i]-excircle[/i] of a triangle $ ABC$ is defined as the circle which touches the segment $ BC$ and the extensions of the segments $ CA$ and $ AB$ beyound the points $ C$ and $ B$, respectively. The center of this circle is the point of intersection of the interior angle bisector of the angle $ CAB$ and the exterior angle bisectors of the angles $ ABC$ and $ BCA$. Similarly, the $ B$-excircle and the $ C$-excircle of triangle $ ABC$ are defined. [hide="Source of the problem"][i]Source of the problem:[/i] Theorem (88) in: John Sturgeon Mackay, [i]The Triangle and its Six Scribed Circles[/i], Proceedings of the Edinburgh Mathematical Society 1 (1883), pages 4-128 and drawings at the end of the volume.[/hide]

Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be two non-concentric circumferences such that $\mathcal{S}_1$ is inside $\mathcal{S}_2$. Let $K$ be a variable point on $\mathcal{S}_1$. The line tangent to $\mathcal{S}_1$ at point $K$ intersects $\mathcal{S}_2$ at points $A$ and $B$. Let $M$ be the midpoint of arc $AB$ that is in the semiplane determined by $AB$ that does not contain $\mathcal{S}_1$. Determine the locus of the point symmetric to $M$ with respect to $K.$

Let $ABC$ be atriangle with $\overline{AC}> \overline{BC}$ and incircle $k$. Let $M,W,L$ be the intersections of the median, angle bisector and altitude from point $C$ respectively. The tangent to $k$ passing through $M$, that is different from $AB$, touch $k$ in $T$. Prove that the angles $\angle MTW$ and $\angle TLM$ are equal.