Let $ABC$ be a triangle with $AB> AC$. Let $D$ be a point on $AB$ such that $DB = DC$ and $M$ the middle of $AC$. The parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC$.

In the following diagram two sides of a square are tangent to a circle with diameter $8$. One corner of the square lies on the circle. There are positive integers $m$ and $n$ so that the area of the square is $m +\sqrt{n}$. Find $m + n$. [asy] import graph; size(4.4cm); real labelscalefactor = 0.5; pen dotstyle = black; filldraw((-0.707106781,0.707106781)--(-0.707106781,-1)--(1,-1)--(1,0.707106781)--cycle,gray, linewidth(1.4)); draw(circle((0,0),1), linewidth(1.4)); [/asy]

Let $ABC$ be a triangle and $D, E$ and $F$ the foots of heights from $A, B$ and $C$ respectively. Let $D_1$ be such a point on $EF$, that $DF = D_1 E$ where $E$ is between $D_1$ and $F$. Similarly, let $D_2$ be such a point on $EF$, that $DE = D_2 F$ where $F$ is between $E$ and $D_2$. Let the bisector of $DD_1$ intersect $AB$ at $P$ and let the bisector of $DD_2$ intersect $AC$ at $Q$. Prove that, $PQ$ bisects $BC$.

Consider the acute-angled triangle $ABC$. Let $X$ be a point on the side $BC$, and $Y$ be a point on the side $CA$. The circle $k_1$ with diameter $AX$ cuts $AC$ again at $E'$ .The circle $k_2$ with diameter $BY$ cuts $BC$ again at $B'$. (i) Let $M$ be the midpoint of $XY$ . Prove that $A'M = B'M$. (ii) Suppose that $k_1$ and $k_2$ meet at $P$ and $Q$. Prove that the orthocentre of $ABC$ lies on the line $PQ$.

$ ABCD$ is a quadrilateral with $ BC$ parallel to $ AD$. $ M$ is the midpoint of $ CD$, $ P$ is the midpoint of $ MA$ and $ Q$ is the midpoint of $ MB$. The lines $ DP$ and $ CQ$ meet at $ N$. Prove that $ N$ is inside the quadrilateral $ ABCD$.

A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle, i.e. both the incircle and circumcircle exists. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.

Consider triangle $ABC$ with $AB<AC<BC$, inscribed in triangle $\Gamma_1$ and the circles $\Gamma_2 (B,AC)$ and $\Gamma_2 (C,AB)$. A common point of circle $\Gamma_2$ and $\Gamma_3$ is point $E$, a common point of circle $\Gamma_1$ and $\Gamma_3$ is point $F$ and a common point of circle $\Gamma_1$ and $\Gamma_2$ is point $G$, where the points $E,F,G$ lie on the same semiplane defined by line $BC$, that point $A$ doesn't lie in. Prove that circumcenter of triangle $EFG$ lies on circle $\Gamma_1$. Note: By notation $\Gamma (K,R)$, we mean random circle $\Gamma$ has center $K$ and radius $R$.

Let $A_1$, $B_1$ and $C_1$ be the midpoints of sides $BC$, $CA$ and $AB$ of triangle $ABC$, respectively. Points $B_2$ and $C_2$ are the midpoints of segments $BA_1$ and $CA_1$ respectively. Point $B_3$ is symmetric to $C_1$ wrt $B$, and $C_3$ is symmetric to $B_1$ wrt $C$. Prove that one of common points of circles $BB_2B_3$ and $CC_2C_3$ lies on the circumcircle of triangle $ABC$.

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

In triangle $ABC$, the midpoints of sides $AB$ and $AC$ are $D$ and $E$, respectively. Prove that the bisectors of the angles $BDE$ and $CED$ intersect at the side $BC$ if the length of side $BC$ is the arithmetic mean of the lengths of sides $AB$ and $AC$.

The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$ [i]Proposed by Michael Ren[/i]

In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$. [i]Proposed by Iman Maghsoudi[/i]

In the triangle $ABC$, $X$ lies on the side $BC$, $Y$ on the side $CA$, and $Z$ on the side $AB$ with $YX \| AB, ZY \| BC$, and $XZ \| CA$. Show that $X,Y$, and $Z$ are the midpoints of the respective sides of $ABC$.

$L$ is a Line that intersect with the side $AB,BC,AC$ of triangle $ABC$ at $F,D,E$ The line perpendicular to $BC$ from $D$ intersect $AB,AC$ at $A_{1},A_{2}$ respectively Name $B_{1},B_{2},C_{1},C_{2}$ similarly Prove that the circumcenters of $AA_{1}A_{2},BB_{1}B_{2},CC_{1}C_{2}$ are collinear

Prove that in a triangle $Sum of medians >\frac{3}{4}(perimeter of triangle )$

Let $O$ and $H$ be the center of the circumcircle and the point of intersection of the altitudes of the acute triangle $ABC$ respectively, $D$ be the foot of the altitude drawn to $BC$, and $E$ be the midpoint of $AO$. Prove that the circumcircle of the triangle $ADE$ passes through the midpoint of the segment $OH$.

Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by Tony Kim[/i]

A [i]synogon [/i] is a convex $2n$-gon with all sides of the same length and all opposite sides are parallel. Show that every synogon can be broken down into a finite number of rhombuses.

The center of a square of side length $ 1$ is placed uniformly at random inside a circle of radius $ 1$. Given that we are allowed to rotate the square about its center, what is the probability that the entire square is contained within the circle for some orientation of the square?

A set of segments with the total length less than $1$ is given on a line. Prove that every set of $n$ points on the line can be translated by a vector of length not exceeding $n/2$, so that all the obtained points are away from the given segments.

A large collection of congruent right triangles is given, each with side length 3,4,5. Find the maximal number of such triangles you can place inside a 20x20 square, with no two triangles intersecting (in their interiors).

In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$? [asy] size(180); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); real r=5/7; pair A=(10,sqrt(28^2-100)),B=origin,C=(20,0),D=(A.x*r,A.y*r); pair bottom=(C.x+(D.x-A.x),C.y+(D.y-A.y)); pair E=extension(D,bottom,B,C); pair top=(E.x+D.x,E.y+D.y); pair F=extension(E,top,A,C); draw(A--B--C--cycle^^D--E--F); dot(A^^B^^C^^D^^E^^F); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,W); label("$E$",E,S); label("$F$",F,dir(0)); [/asy] $\textbf{(A) }48\qquad \textbf{(B) }52\qquad \textbf{(C) }56\qquad \textbf{(D) }60\qquad \textbf{(E) }72\qquad$

Prove that if a lattice triangle has no lattice points on its boundary in addition to its vertices, and one point in its interior, then this interior point is its center of gravity.

Given an acute $\vartriangle ABC$ whose orthocenter is denoted by $H$. A line $\ell$ passes $H$ and intersects $AB,AC$ at $P ,Q$ such that $H$ is the mid-point of $P,Q$. Assume the other intersection of the circumcircle of $\vartriangle ABC$ with the circumcircle of $\vartriangle APQ$ is $X$. Let $C'$ is the symmetric point of $C$ with respect to $X$ and $Y$ is the another intersection of the circumcircle of $\vartriangle ABC$ and $AO$, where O is the circumcenter of $\vartriangle APQ$. Show that $CY$ is tangent to circumcircle of $\vartriangle BCC'$. [img]https://1.bp.blogspot.com/-itG6m1ipAfE/XndLDUtSf7I/AAAAAAAALfc/iZahX6yNItItRSXkDYNofR5hKApyFH84gCK4BGAYYCw/s1600/2018%2Bimoc%2Bg3.png[/img]