Found problems: 1581
Let $ABC$ be an acute-angled triangle and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to the sides $AC$ and $AB$ respectively. Suppose $AD$ is the diameter of the circle $ABC$. Let $M$ be the midpoint of $BC$. Let $K$ be the imsimilicenter of the incircles of the triangles $BMF$ and $CME$. Prove that the points $K, M, D$ are collinear.
[i]Proposed by Bilegdembrel Bat-Amgalan.[/i]
Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon.
[i]Proposed by Hesam Rajabzade[/i]
Let $X,Y,Z,T$ be four points in the plane. The segments $[XY]$ and $[ZT]$ are said to be [i]connected[/i], if there is some point $O$ in the plane such that the triangles $OXY$ and $OZT$ are right-angled at $O$ and isosceles.
Let $ABCDEF$ be a convex hexagon such that the pairs of segments $[AB],[CE],$ and $[BD],[EF]$ are [i]connected[/i]. Show that the points $A,C,D$ and $F$ are the vertices of a parallelogram and $[BC]$ and $[AE]$ are [i]connected[/i].
Let $\Gamma_1$ and $\Gamma_2$ be two circles intersecting at $P$ and $Q$. The common tangent, closer to $P$, of $\Gamma_1$ and $\Gamma_2$ touches $\Gamma_1$ at $A$ and $\Gamma_2$ at $B$. The tangent of $\Gamma_1$ at $P$ meets $\Gamma_2$ at $C$, which is different from $P$, and the extension of $AP$ meets $BC$ at $R$.
Prove that the circumcircle of triangle $PQR$ is tangent to $BP$ and $BR$.
Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$.
([i]Dan Schwarz[/i])
Let $ AB$ be a diameter of a circle, $ C$ be any fixed point between $ A$ and $ B$ on this diameter, and $ Q$ be a variable point on the circumference of the circle. Let $ P$ be the point on the line determined by $ Q$ and $ C$ for which $ \frac{AC}{CB}\equal{}\frac{QC}{CP}$. Describe, with proof, the locus of the point $ P$.
One of the external common tangent lines of the two externally tangent circles with center $O_1$ and $O_2$ touches the circles at $B$ and $C$, respectively. Let $A$ be the common point of the circles. The line $BA$ meets the circle with center $O_2$ at $A$ and $D$. If $|BA|=5$ and $|AD|=4$, then what is $|CD|$?
$ \textbf{(A)}\ \sqrt{20}
\qquad\textbf{(B)}\ \sqrt{27}
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ \frac{15}2
\qquad\textbf{(E)}\ 4\sqrt5
$
Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:
[list=]
[*]$L,$ a rotation of $90^{\circ}$ counterclockwise around the origin;
[*]$R,$ a rotation of $90^{\circ}$ clockwise around the origin;
[*]$H,$ a reflection across the $x$-axis; and
[*]$V,$ a reflection across the $y$-axis.
[/list]
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)
$\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$
Let $ ABC$ be triangle, $ I$ its in-center; $ A_1,B_1,C_1$ be the reflections of $ I$ in $ BC, CA, AB$ respectively. Suppose the circum-circle of triangle $ A_1B_1C_1$ passes through $ A$. Prove that $ B_1,C_1,I,I_1$ are concylic, where $ I_1$ is the in-center of triangle $ A_1,B_1,C_1$.
Let $ABC$ be an acute-angled triangle with circumference $\Omega$. Let the angle bisectors of $\angle B$ and $\angle C$ intersect $\Omega$ again at $M$ and $N$. Let $I$ be the intersection point of these angle bisectors. Let $M'$ and $N'$ be the respective reflections of $M$ and $N$ in $AC$ and $AB$. Prove that the center of the circle passing through $I$, $M'$, $N'$ lies on the altitude of triangle $ABC$ from $A$.
[i]Proposed by Victor Domínguez and Ariel García[/i]
Let $ ABC$ be a triangle and let $ I$ be the incenter. $ Ia$ $ Ib$ and $ Ic$ are the excenters opposite to points $ A$ $ B$ and $ C$ respectively. Let $ La$ be the line joining the orthocenters of triangles $ IBC$ and $ IaBC$. Define $ Lb$ and $ Lc$ in the same way.
Prove that $ La$ $ Lb$ and $ Lc$ are concurrent.
Daniel
Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.
A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors).
Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.)
[hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly.
Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent
Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that:
1) $EY$ is perpendicular to $AD$;
2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.
Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$.
1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$;
2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$.
(Dan Schwarz)
In acute-angled triangle $ABC$, a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides. $r_b$ and $r_c$ are defined similarly. $r$ is the radius of the incircle of $ABC$. Show that \[ \frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}. \]
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?
[asy]
unitsize(10mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--D);
draw(B--G);
draw(C--F);
draw(E--H);
[/asy]
$ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$
Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.
For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be [i]area definite[/i] for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
Consider a circle $C_1$ and a point $O$ on it. Circle $C_2$ with center $O$, intersects $C_1$ in two points $P$ and $Q$. $C_3$ is a circle which is externally tangent to $C_2$ at $R$ and internally tangent to $C_1$ at $S$ and suppose that $RS$ passes through $Q$. Suppose $X$ and $Y$ are second intersection points of $PR$ and $OR$ with $C_1$. Prove that $QX$ is parallel with $SY$.
Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.
A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.