Triangle $ABC$ has $AB=21$, $AC=22$, and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

In an acute-angled triangle $ ABC$, $ D,E,F$ are the feet of the altitudes from $ A,B,C$, respectively, and $ P,Q,R$ are the feet of the perpendiculars from $ A,B,C$ onto $ EF,FD,DE$, respectively. Prove that the lines $ AP,BQ,CR$ are concurrent.

Triangle $ABC$ is given, where $AC<BC$ and $\angle ACB=60^\circ\!\!.$ Point $D$, distinct from $A$, lies on the segment $AC$ such that $AB=BD$, and point $E$, distinct from $B$, lies on the line $BC$ such that $AB=AE$. Prove that $\angle DEC=30^\circ$.

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

Circle $\Omega$ with center $O$ is the circumcircle of an acute triangle $\triangle ABC$ with $AB<BC$ and orthocenter $H$. On the line $BO$ there is point $D$ such that $O$ is between $B$ and $D$ and $\angle ADC= \angle ABC$ . The semi-line starting at $H$ and parallel to $BO$ wich intersects segment $AC$ , intersects $\Omega$ at $E$. Prove that $BH=DE$.

Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$. Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$.

Given is a triangle with side lengths $a,b$ and $c$, incenter $I$ and centroid $S$. Prove: If $a+b=3c$, then $S \neq I$ and line $SI$ is perpendicular to one of the sides of the triangle.

Two points $ B$ and $ C$ are in a plane. Let $ S$ be the set of all points $ A$ in the plane for which $ \triangle ABC$ has area $ 1$. Which of the following describes $ S$? $ \textbf{(A)}\ \text{two parallel lines}\qquad \textbf{(B)}\ \text{a parabola}\qquad \textbf{(C)}\ \text{a circle}\qquad \textbf{(D)}\ \text{a line segment}\qquad \textbf{(E)}\ \text{two points}$

Two regular pentagons $ABCDE$ and $AEKPL$ are given in space, such that $\angle DAK = 60^{\circ}$. Let $M$, $N$ and $S$ be the midpoints of $AE$, $CD$ and $EK$. Prove that: [list=a] [*] $\triangle NMS$ is a right triangle; [*] planes $(ACK)$ and $(BAL)$ are perpendicular. [/list] [i]Ukraine Olympiad[/i]

Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.

A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle? $ \textbf{(A) }160 \qquad \textbf{(B) }180 \qquad \textbf{(C) }222 \qquad \textbf{(D) }228 \qquad \textbf{(E) }390 \qquad $

In $\triangle ABC,$ $AB=6, AC=8, BC=10,$ and $D$ is the midpoint of $\overline{BC}.$ What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC?$ $\textbf{(A)} \sqrt{5} \qquad \textbf{(B)} \frac{11}{4}\qquad \textbf{(C)} 2\sqrt{2} \qquad \textbf{(D)} \frac{17}{6} \qquad \textbf{(E)} 3$

In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.

A robotic lawnmower is located in the middle of a large lawn. Due a manufacturing defect, the robot can only move straight ahead and turn in directions that are multiples of $60^o$. A fence must be set up so that it delimits the entire part of the lawn that the robot can get to, by traveling along a curve with length no more than $10$ meters from its starting position, given that it is facing north when it starts. How long must the fence be?

Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.

The diagram below shows a square of area $36$ separated into two rectangles and a smaller square. One of the rectangles has an area of $12$. What is the smallest rectangle's area? [asy] size(70); draw((0,0)--(2,0)--(2,6)--(0,6)--cycle); draw((2,2)--(6,2)--(6,6)--(2,6)--cycle); draw((2,2)--(6,2)--(6,0)--(2,0)--cycle); label("$12$",(1,3)); label("$?$",(4,4)); label("$?$",(4,1)); [/asy] $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }\text{Not Enough Information}$

A circle with diameter $20$ has points $A, B, C, D, E,$ and $F$ equally spaced along its circumference. A second circle is tangent to the lines $AB$ and $AF$ and internally tangent to the circle. If the second circle has diameter $\sqrt{m}+n$ for integers $m$ and $n$, find $m + n.$ [asy] import geometry; size(180); draw(circle((0,0),5)); pair[] p; string[] l={"A","B","C","D","E","F"}; for (int i=0; i<6; ++i){ p.append(new pair[]{dir(i*60+180)*5}); dot(p[i]); label(l[i],p[i],p[i]/3); } draw(p[0]--p[1]^^p[0]--p[5]); p.append(new pair[]{intersectionpoint(p[0]--p[0]+dir(-60)*90,p[3]--p[3]+(0,-100))}); p.append(new pair[]{intersectionpoint(p[0]--p[0]+dir(+60)*90,p[3]--p[3]+(0,+100))}); draw(incircle(p[0],p[6],p[7]));[/asy]

A cylinder with radius $15$ and height $16$ is inscribed in a sphere. Three congruent smaller spheres of radius $x$ are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of $x$?

$ABC$ is a triangle such that $BC = 10$, $CA = 12$. Let $M$ be the midpoint of side $AC$. Given that $BM$ is parallel to the external bisector of $\angle A$, find area of triangle $ABC$. (Lines $AB$ and $AC$ form two angles, one of which is $\angle BAC$. The external angle bisector of $\angle A$ is the line that bisects the other angle.

Triangle $\vartriangle ABC$ has side lengths $AB = 2$, $CA = 3$ and $BC = 4$. Compute the radius of the circle centered on $BC$ that is tangent to both $AB$ and $AC$.

Show that triangle $ABC$ is right-angled if its angles satisfy the ratio $\cos^2A + \cos ^2B +\ cos ^2C=1$.

Let $ABC$ be a triangle inscribed in the circle $(O)$, with orthocenter $H$. Let d be an arbitrary line which passes through $H$ and intersects $(O)$ at $P$ and $Q$. Draw diameter $AA'$ of circle $(O)$. Lines $A'P$ and $A'Q$ meet $BC$ at $K$ and $L$, respectively. Prove that $O, K, L$ and $A'$ are concyclic.

In a triangle $ABC$, the external bisector of $\angle BAC$ intersects the ray $BC$ at $D$. The feet of the perpendiculars from $B$ and $C$ to line $AD$ are $E$ and $F$, respectively and the foot of the perpendicular from $D$ to $AC$ is $G$. Show that $\angle DGE + \angle DGF = 180^{\circ}$.

In the regular hexagon $ABCDEF$ on the line $AF$, the point $X$ is taken so that the angle $XCD$ is $45^o$. Find the angle $\angle FXE$. (Kiev Olympiad)

In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $ 60^{\circ}$. What is the perimeter of the monster in cm? [asy]size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label("1", (sqrt(3)/4, 1/4), NW); label("$60^\circ$", (1,0)); [/asy] $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ 2\pi \qquad \textbf{(C)}\ \frac53 \pi \qquad \textbf{(D)}\ \frac56 \pi \plus{} 2 \qquad \textbf{(E)}\ \frac53 \pi \plus{} 2$