Found problems: 1581
Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$ and $\mathcal C_3(O_3)$ share a common point and meet again pairwise at the points $A$, $B$ and $C$. Show that if the points $A$, $B$, $C$ are collinear then the points $Q$, $O_1$, $O_2$ and $O_3$ lie on the same circle.
Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.
Let $V$ be a $n-$dimensional vector space over a field $F$ with a basis $\{e_1,e_2, \cdots ,e_n\}$.Prove that for any $m-$dimensional linear subspace $W$ of $V$, the number of elements of the set $W \cap P$ is less than or equal to $2^m$ where $P=\{\lambda_1e_1 + \lambda_2e_2 + \cdots + \lambda_ne_n : \lambda_i=0,1\}$.
A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.
Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q \equal{} \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.
The point $P$ is chosen inside or on the equilateral triangle $ABC$ of side length $1$. The reflection of $P$ with respect to $AB$ is $K$, the reflection of $K$ about $BC$ is $M$, and the reflection of $M$ with respect to $AC$ is $N$. What is the maximum length of $NP$?
$\textbf{(A)}\ 2\sqrt 3\qquad\textbf{(B)}\ \sqrt 3\qquad\textbf{(C)}\ \frac{\sqrt 3}{2} \qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 1$
Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.
Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.
What figure can the central projection of a triangle be? (The center of the projection does not lie on the plane of the triangle.)
In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$, where $a$ appears 2003 times.
Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively.
$(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$.
$(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.
[i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]
a) The incircle of a triangle $ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to the bisector $AL$ in points $Q$ and $P$ respectively. Prove that the lines $PC_0, QB_0$ and $BC$ concur.
b) Let $AL$ be the bisector of a triangle $ABC$. Points $O_1$ and $O_2$ are the circumcenters of triangles $ABL$ and $ACL$ respectively. Points $B_1$ and $C_1$ are the projections of $C$ and $B$ to the bisectors of angles $B$ and $C$ respectively. Prove that the lines $O_1C_1, O_2B_1,$ and $BC$ concur.
c) Prove that the two points obtained in pp. a) and b) coincide.
1-Let $ \triangle ABC$ be a triangle and $ (O)$ its circumcircle. $ D$ is the midpoint of arc $ BC$ which doesn't contain $ A$. We draw a circle $ W$ that is tangent internally to $ (O)$ at $ D$ and tangent to $ BC$.We draw the tangent $ AT$ from $ A$ to circle $ W$.$ P$ is taken on $ AB$ such that $ AP \equal{} AT$.$ P$ and $ T$ are at the same side wrt $ A$.PROVE $ \angle APD \equal{} 90^\circ$.
1. In a triangle $ABC$, $O$ is the circumcenter and $I$ is the incenter. $X$ is the reflection of $I$ to $O$. $A_1$ is foot of the perpendicular from $X$ to $BC$. $B_1$ and $C_1$ are defined similarly. prove that $AA_1$,$BB_1$ and $CC_1$ are concurrent.(12 points)
The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.
The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.
Let $ABC$ be a triangle such that $CA \neq CB$,
the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$,
respectively, and $I$ the incircle.
The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$.
The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$.
Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.
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 $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively.
Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.
Let $ABC$ be a triangle and $D$ be the midpoint of the segment $BC$. The circle that passes through $D$ and tangent to $AB$ at $B$, and the circle that passes through $D$ and tangent to $AC$ at $C$ intersect at $M\neq D$. Let $M'$ be the reflection of $M$ with respect to $BC$. Prove that $M'$ is on $AD$.
Circle $S_1$ has radius $5$. Circle $S_2$ has radius $7$ and has its center lying on $S_1$. Circle $S_3$ has an integer radius and has its center lying on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$?
[i]Ray Li[/i]
In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.
Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$
For $a>2$, let $f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)$ and
let $C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).$ Answer the questions as follows.
(1) Show that the point $(f(t),\ g(t))$ lies on the curve $C$.
(2) Find the normal line of the curve $C$ at the point $\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).$
(3) Let $V(a)$ be the volume of the solid generated by a rotation of the part enclosed by the curve $C$, the nornal line found in (2) and the $x$-axis. Express $V(a)$ in terms of $a$, then find $\lim_{a\to\infty} V(a)$.