$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$.

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

Consider all line segments of length $4$ with one end-point on the line $y = x$ and the other end-point on the line $y = 2x$. Find the equation of the locus of the midpoints of these line segments.

Let $I$ be the incenter of a triangle $ABC$; $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the orthocenters of the triangles $BIC$, $AIC$, $AIB$; $M_{a}$, $M_{b}$, $M_{c}$ be the midpoints of $BC$, $CA$, $AB$, and $S_{a}$, $S_{b}$, $S_{c}$ be the midpoints of $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$. Prove that $M_{a}S_{a}$, $M_{b}S_{b}$, $M_{c}S_{c}$ concur. Proposed by: S Kuznetsov

A square $ ABCD$ with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points $ E,F,G$ where $ E$ is the midpoint of $ BC$, $ F,G$ are on $ AB$ and $ CD$, respectively, and they're positioned that $ AF < FB, DG < GC$ and $ F$ is the directly opposite of $ G$. If $ FB \equal{} x$, the length of the longer parallel side of each trapezoid, find the value of $ x$. [asy]unitsize(2.5cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair[] dotted={(0,0),(0,1),(1,1),(1,0),(1/6,0),(1/6,1),(1/2,1/2),(1,1/2)}; draw(unitsquare); draw((1/6,0)--(1/2,1/2)--(1/6,1)); draw((1/2,1/2)--(1,1/2)); dot(dotted); label("$x$",midpoint((1/6,1)--(1,1)),N);[/asy]$ \displaystyle \textbf{(A)}\ \frac {3}{5} \qquad \textbf{(B)}\ \frac {2}{3} \qquad \textbf{(C)}\ \frac {3}{4} \qquad \textbf{(D)}\ \frac {5}{6} \qquad \textbf{(E)}\ \frac {7}{8}$

Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$ such that $OH$ is parallel to $BC$. Let $AH$ intersects again with the circumcircle of $ABC$ at $X$, and let $XB, XC$ intersect with $OH$ at $Y, Z$, respectively. If the projections of $Y,Z$ to $AB,AC$ are $P,Q$, respectively, show that $PQ$ bisects $BC$. [i]Proposed by usjl[/i]

Construct a quadrilateral given its side lengths and the length of the segment joining the midpoints of its diagonals. (IZ Titovich)

For three points $ X,Y,Z$ let $ R_{XYZ}$ be the circumcircle radius of the triangle $ XYZ.$ If $ ABC$ is a triangle with incircle centre $ I$ then we have: \[ \frac{1}{R_{ABI}} \plus{} \frac{1}{R_{BCI}} \plus{} \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} \plus{} \frac{1}{\bar{BI}} \plus{} \frac{1}{\bar{CI}}.\]

Let $ABC$ be an acute-angled triangle in which $D,E,F$ are points on $BC,CA,AB$ respectively such that $AD \perp BC$;$AE = BC$; and $CF$ bisects $\angle C$ internally, Suppose $CF$ meets $AD$ and $DE$ in $M$ and $N$ respectively. If $FM$$= 2$, $MN =1$, $NC=3$, find the perimeter of $\Delta ABC$.

In triangle $ABC$, $\sin A \sin B \sin C = \frac{1}{1000}$ and $AB \cdot BC \cdot CA = 1000$. What is the area of triangle $ABC$? [i]Proposed by Evan Chen[/i]

The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$

Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$. Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$. [i] (Călin Pop & Vlad Robu) [/i]

Suppose in triangle $ABC$ incircle touches the side $BC$ at $P$ and $\angle APB=\alpha$. Prove that : \[\frac1{p-b}+\frac1{p-c}=\frac2{rtg\alpha}\]

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$. Prove that points $P,Q,P_1$, and $Q_1$ are concyclic. [i]Proposed by Anton Trygub, Ukraine[/i]

Find the maximum possible number of kings on a $12\times 12$ chess table so that each king attacks exactly one of the other kings (a king attacks only the squares that have a common point with the square he sits on).

Jorge's teacher asks him to plot all the ordered pairs $ (w, l)$ of positive integers for which $ w$ is the width and $ l$ is the length of a rectangle with area 12. What should his graph look like? $ \textbf{(A)}$[asy]size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,12)); dot((2,6)); dot((3,4)); dot((4,3)); dot((6,2)); dot((12,1)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy] $ \textbf{(B)}$[asy]size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,1)); dot((3,3)); dot((5,5)); dot((7,7)); dot((9,9)); dot((11,11)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy] $ \textbf{(C)}$[asy]size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,11)); dot((3,9)); dot((5,7)); dot((7,5)); dot((9,3)); dot((11,1)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy] $ \textbf{(D)}$[asy]size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,6)); dot((3,6)); dot((5,6)); dot((7,6)); dot((9,6)); dot((11,6)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy] $ \textbf{(E)}$[asy]size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((6,1)); dot((6,3)); dot((6,5)); dot((6,7)); dot((6,9)); dot((6,11)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]

Let $ABC$ be a triangle with circumcircle $\omega$, circumcenter $O{}$, and orthocenter $H{}$. Let $K{}$ be the midpoint of $AH{}$. The perpendicular to $OK{}$ at $K{}$ intersects $AB{}$ and $AC{}$ at $P{}$ and $Q{}$, respectively. The lines $BK$ and $CK$ intersect $\omega$ again at $X{}$ and $Y{}$, respectively. Prove that the second intersection of the circumcircles of triangles $KPY$ and $KQX$ lies on $\omega$. [i]Stefan Lozanovski[/i]

In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

Consider a positive integer $n$ and two points $A$ and $B$ in a plane. Starting from point $A$, $n$ rays and starting from point $B$, $n$ rays are drawn so that all of them are on the same half-plane defined by the line $AB$ and that the angles formed by the $2n$ rays with the segment $AB$ are all acute. Define circles passing through points $A$, $B$ and each meeting point between the rays. What is the smallest number of [b]distinct [/b] circles that can be defined by this construction?

Let $ABC$ ($BC > AC$) be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $AB$ at the point $D$. The circumcircles of triangles $BCD, OCD$ and $AOB$ intersect the ray $CA$ (beyond $A$) at the points $Q, P$ and $K$, respectively, such that $P \in (AK)$ and $K \in (PQ)$. The line $PD$ intersects the circumcircle of triangle $BKQ$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $BQ$. Prove that $TB = TQ$.

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 $B$.

Andrew Mellon found a piece of melon that is shaped like a octagonal prism where the bases are regular. Upon slicing it in half once, he found that he created a cross-section that is an equilateral hexagon. What is the minimum possible ratio of the height of the melon piece to the side length of the base? [i]Proposed by Lohith Tummala[/i]

In the rectangle $ABCD$, $AB = 2BC$. An equilateral triangle $ABE$ is constructed on the side $AB$ of the rectangle so that its sides $AE$ and $BE$ intersect the segment $CD$. Point $M$ is the midpoint of $BE$. Find the $\angle MCD$.

Points $A, B, C$ and $D$ have these coordinates: $A(3,2), B(3,-2), C(-3,-2)$ and $D(-3, 0)$. The area of quadrilateral $ABCD$ is [asy] for (int i = -4; i <= 4; ++i) { for (int j = -4; j <= 4; ++j) { dot((i,j)); } } draw((0,-4)--(0,4),linewidth(1)); draw((-4,0)--(4,0),linewidth(1)); for (int i = -4; i <= 4; ++i) { draw((i,-1/3)--(i,1/3),linewidth(0.5)); draw((-1/3,i)--(1/3,i),linewidth(0.5)); }[/asy] $ \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 18\qquad\text{(D)}\ 21\qquad\text{(E)}\ 24 $