Construct a triangle given two of its side lengths if it is known that the median drawn from their common vertex divides the angle between them in the ratio $1:2$. (V. Chikin)

Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.

Let $O$ be the circumcenter of the triangle $ABC$. The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$. The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$. The point $W$ is the midpoint of $AZ$. a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$; b) Show that $MW$ is perpendicular to $AY$.

Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. What is $Area(ABCD)/Area(O_1O_2O_3O_4)$? $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac 32 \qquad\textbf{(C)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(D)}\ \dfrac {1+2\sqrt 3}2 \qquad\textbf{(E)}\ \dfrac {1+\sqrt 3}2 $

In the following figure---not drawn to scale!---$E$ is the midpoint of $BC$, triangle $FEC$ has area $7$, and quadrilateral $DBEG$ has area $27$. Triangles $ADG$ and $GEF$ have the same area, $x$. Find $x$. [asy] unitsize(2cm); pair A = (0,38/16); pair B = (0,0); pair C = (38/16,0); pair D = (0,25/16); pair E = (19/16,0); pair F = .4*D+.6*C; draw(D -- C -- B -- A -- E -- F); label("$A$", A, W); label("$B$", B, W); label("$C$", C, S); label("$D$", D, W); label("$E$", E, S); label("$F$", F, N); label("$G$", (17*F-8*C)/9, NE); [/asy]

What is the area of the regular hexagon with perimeter $60$?

Jennifer wants to do origami, and she has a square of side length $ 1$. However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the octagon Jennifer obtains?

If $BK$ is an angle bisector of $\angle ABC$ in triangle $ABC$. Find angles of triangle $ABC$ if $BK=KC=2AK$

In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle $ABC$, the median $m_a$ meets $BC$ at $A'$ and the circumcircle again at $A_1$. The symmedian $s_a$ meets $BC$ at $M$ and the circumcircle again at $A_2$. Given that the line $A_1A_2$ contains the circumcenter $O$ of the triangle, prove that: [i](a) [/i]$\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;$ [i](b) [/i]$1+4b^2c^2 = a^2(b^2+c^2)$

Let $\Gamma_1, \Gamma_2, \Gamma_3$ be three pairwise externally tangent circles with radii $1,2,3,$ respectively. A circle passes through the centers of $\Gamma_2$ and $\Gamma_3$ and is externally tangent to $\Gamma_1$ at a point $P.$ Suppose $A$ and $B$ are the centers of $\Gamma_2$ and $\Gamma_3,$ respectively. What is the value of $\frac{{PA}^2}{{PB}^2}?$ [i]Proposed by Kyle Lee[/i]

Let $ABC$ be an acute angled triangle. Let the altitude from $C$ to $AB$ meet $AB$ at $C'$ and have midpoint $M$, and let the altitude from $B$ to $AC$ meet $AC$ at $B'$ and have midpoint $N$. Let $P$ be the point of intersection of $AM$ and $BB'$ and $Q$ the point of intersection of $AN$ and $CC'$. Prove that the point $M, N, P$ and $Q$ lie on a circle.

How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Suppose that $A$ is a finite subset of $\mathbb{R}^d$ such that (a) every three distinct points in $A$ contain two points that are exactly at unit distance apart, and (b) the Euclidean norm of every point $v$ in $A$ satisfies \[\sqrt{\frac{1}{2}-\frac{1}{2\vert A\vert}} \le \|v\| \le \sqrt{\frac{1}{2}+\frac{1}{2\vert A\vert}}.\] Prove that the cardinality of $A$ is at most $2d+4$.

Let $ABCD$ be a convex quadrilateral, where $M$ is the midpoint of $DC$, $N$ is the midpoint of $BC$, and $O$ is the intersection of diagonals $AC$ and $BD$. Prove that $O$ is the centroid of the triangle $AMN$ if and only if $ABCD$ is a parallelogram.

A triangle $ ABC$ is given with $ \left|AB\right| > \left|AC\right|$. Line $ l$ tangents in a point $ A$ the circumcirle of $ ABC$. A circle centered in $ A$ with radius $ \left|AC\right|$ cuts $ AB$ in the point $ D$ and the line $ l$ in points $ E, F$ (such that $ C$ and $ E$ are in the same halfplane with respect to $ AB$). Prove that the line $ DE$ passes through the incenter of $ ABC$.

In $\triangle ABC$, $AB>AC$, bisector of outer angle $\angle A$ intersects circumcircle of $\triangle ABC$ at $E$. Projection of $E$ on $AB$ is $F$. Prove that $2AF=AB-AC$.

Unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Prove that the sum of perimeters of all the smaller squares having common points with diagonal $AC$ does not exceed 1500. [i]Proposed by A. Kanel-Belov[/i]

The diameters of two circles are $ 8$ inches and $ 12$ inches respectively. The ratio of the area of the smaller to the area of the larger circle is: $ \textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{4}{9} \qquad\textbf{(C)}\ \frac{9}{4} \qquad\textbf{(D)}\ \frac{1}{2} \qquad\textbf{(E)}\ \text{none of these}$

Let $ABC$ be an acute triangle and let $A_1$, $B_1$, $C_1$ be points on the sides $BC, CA$ and $AB$, respectively. Show that the triangles $ABC$ and $A_1B_1C_1$ are similar ($\angle A = \angle A_1, \angle B = \angle B_1,\angle C = \angle C_1$) if and only if the orthocentre of the triangle $A_1B_1C_1$ and the circumcentre of the triangle $ABC$ coincide.

[b]p1.[/b] A permutation on $\{1, 2,..., n\}$ is an ordered arrangement of the numbers. For example, $32154$ is a permutation of $\{1, 2, 3, 4, 5\}$. Does there exist a permutation $a_1a_2... a_n$ of $\{1, 2,..., n\}$ such that $i+a_i$ is a perfect square for every $1 \le i \le n$ when a) $n = 6$ ? b) $n = 13$ ? c) $n = 86$ ? Justify your answers. [b]p2.[/b] Circle $C$ and circle $D$ are tangent at point $P$. Line $L$ is tangent to $C$ at point $Q$ and to $D$ at point $R$ where $Q$ and $R$ are distinct from $P$. Circle $E$ is tangent to $C, D$, and $L$, and lies inside triangle $PQR$. $C$ and $D$ both have radius $8$. Find the radius of $E$, and justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/f/b/4b98367ea64e965369345247fead3456d3d18a.png[/img] [b]p3.[/b] (a) Prove that $\sin 3x = 4 \cos^2 x \sin x - \sin x$ for all real $x$. (b) Prove that $$(4 \cos^2 9^o - 1)(4 \cos^2 27^o - 1)(4 cos^2 81^o - 1)(4 cos^2 243^o - 1)$$ is an integer. [b]p4.[/b] Consider a $3\times 3\times 3$ stack of small cubes making up a large cube (as with the small cubes in a Rubik's cube). An ant crawls on the surface of the large cube to go from one corner of the large cube to the opposite corner. The ant walks only along the edges of the small cubes and covers exactly nine of these edges. How many different paths can the ant take to reach its goal? [b]p5.[/b] Let $m$ and $n$ be positive integers, and consider the rectangular array of points $(i, j)$ with $1 \le i \le m$, $1 \le j \le n$. For what pairs m; n of positive integers does there exist a polygon for which the $mn$ points $(i, j)$ are its vertices, such that each edge is either horizontal or vertical? The figure below depicts such a polygon with $m = 10$, $n = 22$. Thus $10$, $22$ is one such pair. [img]https://cdn.artofproblemsolving.com/attachments/4/5/c76c0fe197a8d1ebef543df8e39114fe9d2078.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle, let ${A}'$, ${B}'$, ${C}'$ be the orthogonal projections of the vertices $A$ ,$B$ ,$C$ on the lines $BC$, $CA$ and $AB$, respectively, and let $X$ be a point on the line $A{A}'$.Let $\gamma_{B}$ be the circle through $B$ and $X$, centred on the line $BC$, and let $\gamma_{C}$ be the circle through $C$ and $X$, centred on the line $BC$.The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$, respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$, respectively.Show that the points $M$, ${M}'$, $N$ and ${N}'$ are collinear.

Let $ C_1,C_2$ and $ C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $ i,j$, we define $ (r_{ij},s_{ij})$ as the two internal tangent lines of $ (C_i,C_j)$. Let $ r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ ABCA'B'C'$. Prove that $ AA',BB'$ and $ CC'$ are concurrent. [img]https://cdn.artofproblemsolving.com/attachments/1/2/5ef098966fc9f48dd06239bc7ee803ce4701e2.png[/img]

The circle $\omega_2$ passing through the center $O$ of the circle $\omega_1$, is tangent to the circle $\omega_2$ at the point $A$. On the circle $\omega_2$, the point $C$ is taken so that the ray $AC$ intersects the circle $\omega_1$ for second time at point $D$, the ray $OC$ intersects the circle $\omega_1$ at point $E$ and the lines $DE$ and $AO$ are parallel. Find the size of the angle $DAE$.

A circle $\omega$ touched lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle meets $a$ and $b$ at $X$ and $Y$ respectively. Points $X'$ and $Y'$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Find the locus of projections of the center of the circle to the lines $X'Y'$.

The quadrilateral $ABCD$ is circumscribed about the circle, touches the sides $AB, BC, CD, DA$ in the points $K, L, M, N,$ respectively. Let $P, Q, R, S$ midpoints of the sides $KL, LM, MN, NK$. Prove that $PR = QS$ if and only if $ABCD$ is inscribed.