Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

Let $ABC$ be a triangle. On its sides $AB$ and $BC$ are fixed points $C_1$ and $A_1$, respectively. Find a point $ P$ on the circumscribed circle of triangle $ABC$ such that the distance between the centers of the circumscribed circles of the triangles $APC_1$ and $CPA_1$ is minimal.

Let $E$ be a point outside the square $ABCD$ such that $m(\widehat{BEC})=90^{\circ}$, $F\in [CE]$, $[AF]\perp [CE]$, $|AB|=25$, and $|BE|=7$. What is $|AF|$? $ \textbf{(A)}\ 29 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 31 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 33 $

Let $ABC$ be a acute-angle triangle and $X$ be point in the plane of this triangle. Let $M,N,P$ be the orthogonal projections of $X$ in the lines that contains the altitudes of this triangle Determine the positions of the point $X$ such that the triangle $MNP$ is congruent to $ABC$

A point $O$ lies inside $\triangle ABC$ so that $\angle BOC=90-\angle BAC$. Let $BO, CO$ meet $AC, AB$ at $K, L$. Points $K_1, L_1$ lie on the segments $CL, BK$ so that $K_1B=K_1K$ and $L_1C=L_1L$. If $M$ is the midpoint of $BC$, then prove that $\angle K_1ML_1=90^{o}$. [i]Proposed by Anton Trygub[/i]

Points $D$ and $E$ are taken on sides $BC$ and $CA$ of a triangle $ BD\equal{}AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of $\angle ACB$ intersects $AD$ and $BE$ at $Q$ and $R$ respectively. Prove that $ \frac{PQ}{PR}\equal{}\frac{AD}{BE}$.

Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the ratio of the length of each diagonal to the length of the corresponding parallel side is the same, and find this ratio.

$\boxed{\text{G1}}$ Let $M$ be interior point of the triangle $ABC$ with <BAC=70and <ABC=80 If <ACM=10 and <CBM=20.Prove that $AB=MC$

Let $ABCDS$ be a (not neccessarily straight) pyramid with a rectangular base $ABCD$ and acute triangular faces $ABS,BCS,CDS,DAS$. We consider all cuboids which are inscribed inside the pyramid with its base being in the plane $ABCD$ and its upper vertexes are in the triangular faces (one in each). Find the locus of the midpoints of these cuboids.

Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$.

The midpoints of the sides $BC$, $CA$ and $AB$ of triangle $ABC$ are $M$, $N$ and $P$ respectively . $G$ is the intersection point of the medians. The circumscribed circle around $BGP$ intersects the line $MP$ at the point $K$ (different than $P$).The circle circumscribed around $CGN$ intersects the line $MN$ at point $L$ (different than $N$). Prove that $\angle BAK = \angle CAL$.

Triangle $ABC$ lies entirely inside a polygon. Prove that the perimeter of triangle $ABC$ is not greater than that of the polygon.

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\vartriangle MNP$ can be written in simplest form as $a/b$. Find $a + b$.

Let $A, B,C$ be colinear points in this order, $\omega$ an arbitrary circle passing through $B$ and $C$, and $l$ an arbitrary line different from $BC$, passing through A and intersecting $\omega$ at $M$ and $N$. The bisectors of the angles $\angle CMB$ and $\angle CNB$ intersect $BC$ at $P$ and $Q$. Prove that $AP\cdot AQ = AB \cdot AC$.

Triangle $ABC$ is given for which $BC = AC + \frac{1}{2}AB$. The point $P$ divides $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2\angle CPA$.

Acute-angled $~$ $\triangle{ABC}$ $~$ is given. A line $~$ $l$ $~$ parallel to side $~$ $AB$ $~$ passing through vertex $~$ $C$ $~$ is drawn. Let the angle bisectors of $~$ $\angle{BAC}$ $~$ and $~$ $\angle{ABC}$ $~$ intersect the sides $~$ $BC$ and $~$ $AC$ at points $~$ $D$ $~$ and $~$ $F$, and line $~$ $l$ $~$ at points $~$ $E$ $~$ and $~$ $G$ $~$ respectively. Prove that if $~$ $\overline{DE}=\overline{GF}$ $~$ then $~$ $\overline{AC}=\overline{BC}\, .$

Three numbers have sum $k$ (where $k\in \mathbb{R}$) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add $\frac{k}{6}$ than we have geometry progression. Find those numbers?

A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.

Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$ a) Prove that $A,P,Q$ are collinear. b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.

Consider $\triangle \natural\flat\sharp$. Let $\flat\sharp$, $\sharp\natural$ and $\natural\flat$ be the answers to problems $4$, $5$, and $6$, respectively. If the incircle of $\triangle \natural\flat\sharp$ touches $\natural\flat$ at $\odot$, find $\flat\odot$. [i]Proposed by Evan Chen[/i]

The eight pointed star is a popular quilting pattern. What percent of the entire 4-by-4 grid is covered by the star? $(A)40$ $~~~$ $(B)50$ $~~~$ $(C)60$ $~~~$ $(D)75$ $~~~$ $(E)80$

In non-isosceles acute ${}{\triangle ABC}$, $AP$, $BQ$, $CR$ is the height of the triangle. $A_1$ is the midpoint of $BC$, $AA_1$ intersects $QR$ at $K$, $QR$ intersects a straight line that crosses ${A}$ and is parallel to $BC$ at point ${D}$, the line connecting the midpoint of $AH$ and ${K}$ intersects $DA_1$ at $A_2$. Similarly define $B_2$, $C_2$. ${}\triangle A_2B_2C_2$ is known to be non-degenerate, and its circumscribed circle is $\omega$. Prove that: there are circles $\odot A'$, $\odot B'$, $\odot C'$ tangent to and INSIDE $\omega$ satisfying: (1) $\odot A'$ is tangent to $AB$ and $AC$, $\odot B'$ is tangent to $BC$ and $BA$, and $\odot C'$ is tangent to $CA$ and $CB$. (2) $A'$, $B'$, $C$' are different and collinear. [i]Created by Sihui Zhang[/i]

Five spheres of radius $r$ are inside a right circular cone. Four of the spheres lie on the base of the cone. Each touches two of the others and the sloping sides of the cone. The fifth sphere touches each of the other four and also the sloping sides of the cone. Find the volume of the cone.

Let $ABC$ be a triangle and let $\Omega$ be its circumcircle. The internal bisectors of angles $A, B$ and $C$ intersect $\Omega$ at $A_1, B_1$ and $C_1$, respectively, and the internal bisectors of angles $A_1, B_1$ and $C_1$ of the triangles $A_1 A_2 A_ 3$ intersect $\Omega$ at $A_2, B_2$ and $C_2$, respectively. If the smallest angle of the triangle $ABC$ is $40^{\circ}$, what is the magnitude of the smallest angle of the triangle $A_2 B_2 C_2$ in degrees?