Found problems: 25757
2020 CHMMC Winter (2020-21), 12
Let $\Omega_1$ and $\Omega_2$ be two circles intersecting at distinct points $P$ and $Q$. The line tangent to $\Omega_1$ at $P$ passes through $\Omega_2$ at a second point $A$, and the line tangent to $\Omega_2$ at $P$ passes through $\Omega_1$ at a second point $B$. Ray $AQ$ intersects $\Omega_1$ at a second point $C$, and ray $BQ$ intersects $\Omega_2$ at a second point $D$. Suppose that $\angle CPD > \angle APB$ (measuring both angles as the non-reflex angle) and that
\[
\frac{\text{Area}(CPD)}{PA \cdot PB} = \frac{1}{4}.
\]
Find the sum of all possible measures of $\angle APB$ in degrees.
2021 Bulgaria National Olympiad, 2
A point $T$ is given on the altitude through point $C$ in the acute triangle $ABC$ with circumcenter $O$, such that $\measuredangle TBA=\measuredangle ACB$. If the line $CO$ intersects side $AB$ at point $K$, prove that the perpendicular bisector of $AB$, the altitude through $A$ and the segment $KT$ are concurrent.
2020 LIMIT Category 1, 13
On the side $AC$ of an acute triangle $\triangle ABC$, a point $D$ is taken such that $2AD=CD=2, BD\perp AC$. A circle of radius $2$ passes through $A$ and $D$ and is tangent to the circumcircle of $\triangle BDC$. Find $[\text{Area}(\triangle ABC)]$ where $[.]$ is the greatest integer function.
2003 Cono Sur Olympiad, 3
Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.
2004 Tournament Of Towns, 4
A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.
2022 Oral Moscow Geometry Olympiad, 5
Circle $\omega$ is tangent to the interior of the circle $\Omega$ at the point C. Chord $AB$ of circle $\Omega$ is tangent to $\omega$. Chords $CF$ and $BG$ of circle $\Omega$ intersect at point $E$ lying on $\omega$. Prove that the circumcircle of triangle $CGE$ is tangent to straight line $AF$.
(I. Kukharchuk)
2004 Greece National Olympiad, 3
Consider a circle $K(O,r)$ and a point $A$ outside $K.$ A line $\epsilon$ different from $AO$ cuts $K$ at $B$ and $C,$ where $B$ lies between $A$ and $C.$
Now the symmetric line of $\epsilon$ with respect to axis of symmetry the line $AO$ cuts $K$ at $E$ and $D,$ where $E$ lies between $A$ and $D.$
Show that the diagonals of the quadrilateral $BCDE$ intersect in a fixed point.
KoMaL A Problems 2020/2021, A. 796
Let $ABCD$ be a cyclic quadrilateral. Let lines $AB$ and $CD$ intersect in $P,$ and lines $BC$ and $DA$ intersect in $Q.$ The feet of the perpendiculars from $P$ to $BC$ and $DA$ are $K$ and $L,$ and the feet of the perpendiculars from $Q$ to $AB$ and $CD$ are $M$ and $N.$ The midpoint of diagonal $AC$ is $F.$
Prove that the circumcircles of triangles $FKN$ and $FLM,$ and the line $PQ$ are concurrent.
[i]Based on a problem by Ádám Péter Balogh, Szeged[/i]
1997 Czech and Slovak Match, 4
Is it possible to place $100$ balls in space so that no two of them have a common interior point and each of them touches at least one third of the others?
2020 LIMIT Category 2, 11
$\triangle PQR$ is isosceles and right angled at $R$. Point $A$ is inside $\triangle PQR$, such that $PA=11, QA=7$, and $RA=6$. Legs $\overline{PR}$ and $\overline{QR}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?
2021 JBMO Shortlist, G5
Let $ABC$ be an acute scalene triangle with circumcircle $\omega$. Let $P$ and $Q$ be interior points of the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$. Let $L$ be a point on $\omega$ such that $AL$ is parallel to $BC$. The segments $BQ$ and $CP$ intersect at $S$. The line $LS$ intersects $\omega$ at $K$. Prove that $\angle BKP = \angle CKQ$.
Proposed by [i]Ervin Macić, Bosnia and Herzegovina[/i]
2001 National Olympiad First Round, 29
Let $ABCD$ be a isosceles trapezoid such that $AB || CD$ and all of its sides are tangent to a circle. $[AD]$ touches this circle at $N$. $NC$ and $NB$ meet the circle again at $K$ and $L$, respectively. What is $\dfrac {|BN|}{|BL|} + \dfrac {|CN|}{|CK|}$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 9
\qquad\textbf{(E)}\ 10
$
May Olympiad L1 - geometry, 2001.2
Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm.
We do three folds:
1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$.
A right trapezoid $BCDQ$ is then formed.
2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed.
3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$.
After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$.
Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.
1994 Spain Mathematical Olympiad, 5
Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.
2017 South Africa National Olympiad, 2
Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$. $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$, determine the length of $XY$.
2011 Federal Competition For Advanced Students, Part 2, 3
Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$.
We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$. ($PBQA$ is a convex quadrangle.)
Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$.
The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ .
Show that there is a point $Z$ that lies on all of these lines $UV$.
Brazil L2 Finals (OBM) - geometry, 2019.3
Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the
vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre
1970 IMO Longlists, 40
Let ABC be a triangle with angles $\alpha, \beta, \gamma$ commensurable with $\pi$. Starting from a point $P$ interior to the triangle, a ball reflects on the sides of $ABC$, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices $A,B,C$, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment $0$ to infinity consists of segments parallel to a finite set of lines.
2007 Korea National Olympiad, 2
$ ABC$ is a triangle which is not isosceles. Let the circumcenter and orthocenter of $ ABC$ be $ O$, $ H$, respectively,
and the altitudes of $ ABC$ be $ AD$, $ BC$, $ CF$. Let $ K\neq A$ be the intersection of $ AD$ and circumcircle of triangle $ ABC$, $ L$ be the intersection of $ OK$ and $ BC$, $ M$ be the midpoint of $ BC$, $ P$ be the intersection of $ AM$ and the line that passes $ L$ and perpendicular to $ BC$, $ Q$ be the intersection of $ AD$ and the line that passes $ P$ and parallel to $ MH$, $ R$ be the intersection of line $ EQ$ and $ AB$, $ S$ be the intersection of $ FD$ and $ BE$.
If $ OL \equal{} KL$, then prove that two lines $ OH$ and $ RS$ are orthogonal.
2020 Thailand TST, 1
Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $\omega_0$ be a circle tangent to chord $AB$ and arc $ACB$. For each $i = 1, 2$, let $\omega_i$ be a circle tangent to $AB$ at $T_i$ , to $\omega_0$ at $S_i$ , and to arc $ACB$. Suppose $\omega_1 \ne \omega_2$. Prove that there is a circle passing through $S_1, S_2, T_1$, and $T_2$, and tangent to $\Gamma$ if and only if $\angle ACB = 90^o$.
.
2018 USAJMO, 3
Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.
2009 Jozsef Wildt International Math Competition, w. 24
If $K$, $L$, $M$ denote the midpoints of the sides $AB$, $BC$, $CA$ in triangle $\triangle ABC$, then for all $P$ in the plane of triangle $\triangle ABC$, we have $$\frac{AB}{PK}+\frac{BC}{PL}+\frac{CA}{PM} \geq \frac{AB\cdot BC \cdot CA}{4\cdot PK\cdot PL\cdot PM}$$
Indonesia MO Shortlist - geometry, g1.1
$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.
1996 AIME Problems, 7
Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?
2008 Chile National Olympiad, 4
Three colors are available to paint the plane. If each point in the plane is assigned one of these three colors, prove that there is a segment of length $1$ whose endpoints have the same color.