About the triangle $ABC$ it is known that $AM$ is its median, and $\angle AMC = \angle BAC$. On the ray $AM$ lies the point $K$ such that $\angle ACK = \angle BAC$. Prove that the centers of the circumcircles of the triangles $ABC, ABM$ and $KCM$ lie on the same line.

Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.

A triangle $ABC$ is given. A circle $\gamma$ centered at $A$ meets segments $AB$ and $AC$. The common chord of $\gamma$ and the circumcircle of $ABC$ meets $AB$ and $AC$ at $X$ and $Y$, respectively. The segments $CX$ and $BY$ meet $\gamma$ at point $S$ and $T$, respectively. The circumcircles of triangles $ACT$ and $BAS$ meet at points $A$ and $P$. Prove that $CX, BY$ and $AP$ concur.

Let ${ABC}$ be a triangle inscribed in a circle. Point ${P}$ is the center of the arc ${BAC}$. The circle with the diameter ${CP}$ intersects the angle bisector of angle ${\angle BAC}$ at points ${K, L}$ ${(|AK| <|AL|)}$. Point ${M}$ is the reflection of ${L}$ with respect to line ${BC}$. Prove that the circumcircle of the triangle ${BKM}$ passes through the center of the segment ${BC}$ .

In triangle $ABC$ with $AB = 26$, $BC = 28$, and $C A = 30$, let $M$ be the midpoint of $AB$ and let $N$ be the midpoint of $C A$. The circumcircle of triangle $BCM$ intersects $AC$ at $X\ne C$, and the circumcircle of triangle $BCN $intersects $AB$ at $Y\ne B$. Lines $MX$ and $NY$ intersect $BC$ at $P$ and $Q$, respectively. The area of quadrilateral $PQY X$ can be expressed as $\frac{p}{q}$ for positive integers $p$ and $q$ such that gcd$(p,q) = 1$. Find $q$.

Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.

In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.

Let $ABCD$ be a parallelogram. $E$ and $F$ are on $BC, CD$ respectively such that the triangles $ABE$ and $BCF$ have the same area. Let $BD$ intersect $AE, AF$ at $M, N$ respectively. Prove there exists a triangle whose side lengths are $BM, MN, ND$.

Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.

p1. From the measurement of the height of nine trees obtained data as following. a) There are three different measurement results (in meters) b) All data are positive numbers c) Mean$ =$ median $=$ mode $= 3$ d) The sum of the squares of all data is $87.$ Determine all possible heights of the nine trees. p2. If $x$ and $y$ are integers, find the number of pairs $(x,y)$ that satisfy $|x|+|y|\le 50$. p3. The plane figure $ABCD$ on the side is a trapezoid with $AB$ parallel to $CD$. Points $E$ and $F$ lie on $CD$ so that $AD$ is parallel to $BE$ and $AF$ is parallel to $BC$. Point $H$ is the intersection of $AF$ with $BE$ and point $G$ is the intersection of $AC$ with $BE$. If the length of $AB$ is $4$ cm and the length of $CD$ is $10$ cm, calculate the ratio of the area of ​​the triangle $AGH$ to the area of ​​the trapezoid $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/c/7/e751fa791bce62f091024932c73672a518a240.png[/img] p4. A prospective doctor is required to intern in a hospital for five days in July $2011$. The hospital leadership gave the following rules: a) Internships may not be conducted on two consecutive days. b) The fifth day of internship can only be done after four days counted since the fourth day of internship. Suppose the fourth day of internship is the date $20$, then the fifth day of internship can only be carried out at least the date $24$. Determine the many possible schedule options for the prospective doctor. p5. Consider the following sequences of natural numbers: $5$, $55$, $555$, $5555$, $55555$, $...$ ,$\underbrace{\hbox{5555...555555...}}_{\hbox{n\,\,numbers}}$ . The above sequence has a rule: the $n$th term consists of $n$ numbers (digits) $5$. Show that any of the terms of the sequence is divisible by $2011$.

Let $ n$ be the smallest positive integer such that $ n$ is divisible by $ 20$, $ n^2$ is a perfect cube, and $ n^3$ is a perfect square. What is the number of digits of $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

Let $ABC$ be an acute-angled triangle. In $ABC$, $AB \neq AB$, $K$ is the midpoint of the the median $AD$, $DE \perp AB$ at $E$, $DF \perp AC$ at $F$. The lines $KE$, $KF$ intersect the line $BC$ at $M$, $N$, respectively. The circumcenters of $\triangle DEM$, $\triangle DFN$ are $O_1, O_2$, respectively. Prove that $O_1 O_2 \parallel BC$.

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

In $\triangle{ABC}$ with side lengths $AB = 13$, $AC = 12$, and $BC = 5$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle{ABC}$. What is the area of $\triangle{MOI}$? $\textbf{(A)}\ 5/2\qquad\textbf{(B)}\ 11/4\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 13/4\qquad\textbf{(E)}\ 7/2$

We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear. [i]Proposed by Sammy Luo[/i]

$w_B$ and $w_C$ are excircles of a triangle $ABC$. The circle $w_B'$ is symmetric to $w_B$ with respect to the midpoint of $AC$, the circle $w_C'$ is symmetric to $w_C$ with respect to the midpoint of $AB$. Prove that the radical axis of $w_B'$ and $w_C'$ halves the perimeter of $ABC$.

Let $A,B,C,D,E,F$ be the midpoints of consecutive sides of a hexagon with parallel opposite sides. Prove that the points $AB\cap DE$, $BC\cap EF$, $AC\cap DF$ lie on a line.

Each face of a cube has been divided into four equal quarters and each quarter is painted with one of three available colours. Quarters with common sides are painted with different colours . Prove that each of the available colours was used in painting $8$ quarters.

Two circles intersect at $A$ and $B$. A line through $A$ meets the first circle again at $C$ and the second circle again at $D$. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ not containing $A$, and let $K$ be the midpoint of the segment $CD$. Show that $\angle MKN =\pi/2$. (You may assume that $C$ and $D$ lie on opposite sides of $A$.) [i]D. Tereshin[/i]

Given an acute and scalene triangle $ABC$ with $AB<AC$ and random line $(e)$ that passes throuh the center of the circumscribed circles $c(O,R)$. Line $(e)$, intersects sides $BC,AC,AB$ at points $A_1,B_1,C_1$ respectively (point $C_1$ lies on the extension of $AB$ towards $B$). Perpendicular from $A$ on line $(e)$ and $AA_1$ intersect circumscribed circle $c(O,R)$ at points $M$ and $A_2$ respectively. Prove that a) points $O,A_1,A_2, M$ are consyclic b) if $(c_2)$ is the circumcircle of triangle $(OBC_1)$ and $(c_3)$ is the circumcircle of triangle $(OCB_1)$, then circles $(c_1),(c_2)$ and $(c_3)$ have a common chord

Let $ABC$ be an equilateral triangle, and $P$ is a point on its incircle. Let $\omega_a$ be the circle tangent to $AB$ passing through $P$ and $A$. Similarly, let $\omega_b$ be the circle tangent to $BC$ passing through $P$ and $B$, and $\omega_c$ be the circle tangent to $CA$ passing through $P$ and $C$. Prove that the circles $\omega_a$, $\omega_b$, $\omega_c$ has a common tangent line. [i]Proposed by Ivan Chan Kai Chin[/i]

Assuming that each point of a straight line is painted red or blue, arbitrarily, show that it is always possible to choose three points $A, B$ and $C$ in such a way straight, that are painted the same color and that: $$\frac{AB}{1}=\frac{BC}{2}=\frac{AC}{3}.$$

Suppose that $ABCD$ is a parallelogram such that $\angle DAC = 90^o$. Let $H$ be the foot of perpendicular from $A$ to $DC$, also let $P$ be a point along the line $AC$ such that the line $PD$ is tangent to the circumcircle of the triangle $ABD$. Prove that $\angle PBA = \angle DBH$. Proposed by Iman Maghsoudi

Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$.

Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Prove that $AA'$, $BB'$ and $CC'$ are concurrent.