Found problems: 25757
2025 Israel TST, P2
Given a cyclic quadrilateral $ABCD$, define $E$ as $AD \cap BC$ and $F$ as $AB \cap CD$.
Let $\Omega_A$ be the circle passing through $A, D$ and tangent to $AB$, and let its center be $O_A$.
Let $\Gamma_B$ be the circle passing through $B, C$ and tangent to $AB$, and let its center be $O_B$.
Let $\Gamma_C$ be the circle passing through $B, C$ and tangent to $CD$, and let its center be $O_C$.
Let $\Omega_D$ be the circle passing through $A, D$ and tangent to $CD$, and let its center be $O_D$.
Prove that $O_AO_BO_CO_D$ is cyclic, and prove that it's center lies on $EF$.
2024 Malaysia IMONST 2, 6
Rui Xuen has a circle $\omega$ with center $O$, and a square $ABCJ$ with vertices on $\omega$. Let $M$ be the midpoint of $AB$, and let $\Gamma$ be the circle passing through the points $J$, $O$, $M$. Suppose $\Gamma$ intersect line $AJ$ at a point $P \neq J$, and suppose $\Gamma$ intersect $\omega$ at a point $Q \neq J$. A point $R$ lies on side $BC$ so that $RC = 3RB$.
Help Rui Xuen prove that the points $P$, $Q$, $R$ are collinear.
2022 Kosovo National Mathematical Olympiad, 3
Let $\bigtriangleup ABC$ be a triangle and $D$ be a point in line $BC$ such that $AD$ bisects $\angle BAC$. Furthermore, let $F$ and $G$ be points on the circumcircle of $\bigtriangleup ABC$ and $E\neq D$ point in line $BC$ such that $AF=AE=AD=AG$. If $X$ and $Y$ are the feet of perpendiculars from $D$ to $EF$ and $EG,$ respectively. Prove that $XY\parallel AD$.
2005 Thailand Mathematical Olympiad, 1
Let $ABCD$ be a trapezoid inscribed in a unit circle with diameter $AB$. If $DC = 4AD$, compute $AD$.
1963 Poland - Second Round, 4
In the triangle $ ABC $, the bisectors of the internal and external angles are drawn at the vertices $ A $ and $ B $. Prove that the orthogonal projections of the point $ C $ on these bisectors lie on one straight line.
2019 Sharygin Geometry Olympiad, 13
Let $ABC$ be an acute-angled triangle with altitude $AT = h$. The line passing through its circumcenter $O$ and incenter $I$ meets the sides $AB$ and $AC$ at points $F$ and $N$, respectively. It is known that $BFNC$ is a cyclic quadrilateral. Find the sum of the distances from the orthocenter of $ABC$ to its vertices.
2001 Kazakhstan National Olympiad, 8
There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.
2001 Moldova National Olympiad, Problem 3
In a triangle $ABC$, the line symmetric to the median through $A$ with respect to the bisector of the angle at $A$ intersects $BC$ at $M$. Points $P$ on $AB$ and $Q$ on $AC$ are chosen such that $MP\parallel AC$ and $MQ\parallel AB$. Prove that the circumcircle of the triangle $MPQ$ is tangent to the line $BC$.
2021 IOM, 2
Points $P$ and $Q$ are chosen on the side $BC$ of triangle $ABC$ so that $P$ lies between $B$ and $Q$. The rays $AP$ and $AQ$ divide the angle $BAC$ into three equal parts. It is known that the triangle $APQ$ is acute-angled. Denote by $B_1,P_1,Q_1,C_1$ the projections of points $B,P,Q,C$ onto the lines $AP,AQ,AP,AQ$, respectively. Prove that lines $B_1P_1$ and $C_1Q_1$ meet on line $BC$.
2016 PUMaC Geometry A, 7
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \ne A$. If $AB = 10, AD = 24, XA = 17$, and $XB = 21$, then the area of $\vartriangle DEY$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.
2019-IMOC, G5
Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$. The exterior angle bisector of $\angle BAC$ intersects circumcircle of $\vartriangle ABC$ at $N \ne A$. Let $D$ be another intersection of $HN$ and the circumcircle of $\vartriangle ABC$. The line passing through $O$, which is parallel to $AN$, intersects $AB,AC$ at $E, F$, respectively. Prove that $DH$ bisects the angle $\angle EDF$.
[img]https://3.bp.blogspot.com/-F1mFwojG_I0/XnYNR8ofqSI/AAAAAAAALeo/zge24WF0EO8umPAaXprKAeXJHAj7pr6tQCK4BGAYYCw/s1600/imoc2019g5.png[/img]
2013 Princeton University Math Competition, 8
Eight all different sushis are placed evenly on the edge of a round table, whose surface can rotate around the center. Eight people also evenly sit around the table, each with one sushi in front. Each person has one favorite sushi among these eight, and they are all distinct. They find that no matter how they rotate the table, there are never more than three people who have their favorite sushis in front of them simultaneously. By this requirement, how many different possible arrangements of the eight sushis are there? Two arrangements that differ by a rotation are considered the same.
2020 Azerbaijan IZHO TST, 2
Consider two circles $k_1,k_2$ touching at point $T$.
A line touches $k_2$ at point $X$ and intersects $k_1$ at points $A,B$ where $B$ lies between $A$ and $X$.Let $S$ be the second intersection point of $k_1$ with $XT$. On the arc $\overarc{TS}$ not containing $A$ and $B$ , a point $C$ is choosen.
Let $CY$ be the tangent line to $k_2$ with $Y\in{k_2}$ , such that the segment $CY$ doesn't intersect the segment $ST$ .If $I=XY\cap{SC}$ , prove that :
$(a)$ the points $C,T,Y,I$ are concyclic.
$(b)$ $I$ is the $A-excenter$ of $\triangle ABC$
2005 MOP Homework, 3
Let $M$ be the midpoint of side $BC$ of triangle $ABC$ ($AB>AC$), and let $AL$ be the bisector of the angle $A$. The line passing through $M$ perpendicular to $AL$ intersects the side $AB$ at the point $D$. Prove that $AD+MC$ is equal to half the perimeter of triangle $ABC$.
2019 Romania National Olympiad, 2
Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that:
a) $ND = PC$
b) $ND\perp PC$.
2010 CHKMO, 3
Let $ \triangle ABC$ be a right-angled triangle with $ \angle C\equal{}90^\circ$. $ CD$ is the altitude from $ C$ to $ AB$, with $ D$ on $ AB$. $ \omega$ is the circumcircle of $ \triangle BCD$. $ \omega_1$ is a circle situated in $ \triangle ACD$, it is tangent to the segments $ AD$ and $ AC$ at $ M$ and $ N$ respectively, and is also tangent to circle $ \omega$.
(i) Show that $ BD\cdot CN\plus{}BC\cdot DM\equal{}CD\cdot BM$.
(ii) Show that $ BM\equal{}BC$.
1979 AMC 12/AHSME, 1
[asy]
draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle);
draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle);
label("$F$",(0,0),E);
label("$A$",(-2,1),W);
label("$B$",(2,1),E);
label("$C$", (2,-1),E);
label("$D$",(-2,-1),WSW);
label("$E$",(-2,0),W);
label("$G$",(0,-1),S);
//Credit to TheMaskedMagician for the diagram
[/asy]
If rectangle $ABCD$ has area $72$ square meters and $E$ and $G$ are the midpoints of sides $AD$ and $CD$, respectively, then the area of rectangle $DEFG$ in square meters is
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
2009 Indonesia TST, 1
Given an $ n\times n$ chessboard.
a) Find the number of rectangles on the chessboard.
b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?
1963 All Russian Mathematical Olympiad, 031
Given two fixed points $A$ and $B$ .The point $M$ runs along the circumference containing $A$ and $B$. $K$ is the midpoint of the segment $[MB]$. $[KP]$ is a perpendicular to the line $(MA)$.
a) Prove that all the possible lines $(KP)$ pass through one point.
b) Find the set of all the possible points $P$.
2002 Chile National Olympiad, 3
Given the line $AB$, let $M$ be a point on it. Towards the same side of the plane and with bases $AM$ and $MB$, squares $AMCD$ and $MBEF$ are constructed. Let $N$ be the point (different from $M$) where the circumcircles circumscribed to both squares intersect and let $N_1$ be the point where the lines $BC$ and $AF$ intersect. Prove that the points $N$ and $N_1$ coincide. Prove that as the point $M$ moves on the line $AB$, the line $MN$ moves always passing through a fixed point.
2024 Sharygin Geometry Olympiad, 8.7
A convex quadrilateral $ABCD$ is given. A line $l \parallel AC$ meets the lines $AD$, $BC$, $AB$, $CD$ at points $X$, $Y$, $Z$, $T$ respectively. The circumcircles of triangles $XYB$ and $ZTB$ meet for the second time at point $R$. Prove that $R$ lies on $BD$.
2024 Serbia National Math Olympiad, 4
Let $ABC$ be a triangle with incenter and $A$-excenter $I, I_a$, whose incircle touches $BC, CA, AB$ at $D, E, F$. The line $EF$ meets $BC$ at $P$ and $X$ is the midpoint of $PD$. Show that $XI \perp DI_a$.
1963 Dutch Mathematical Olympiad, 2
The straight lines $k$ and $\ell$ intersect at right angles. A line intersects $k$ in $A$ and $\ell$ in $B$. Consider all straight line segments $PQ$ ($P$ on $k$ and $Q$ on $\ell$), which makes an angle of $45^o$ with $AB$.
(a) Determine the locus of the midpoints of the line segments $PQ$,
(b) If the perpendicular bisector of such a line segment $PQ$ intersects the line $k$ at $K$ and the line $\ell$ at $L$, then prove that $KL \ge PQ$.
[hide=original wording of second sentence]De loodrechte snijlijn van k en l snijdt k in A en t in B[/hide]
2020 Purple Comet Problems, 5
The diagram below shows square $ABCD$ which has side length $12$ and has the same center as square $EFGH$ which has side length $6$. Find the area of quadrilateral $ABFE$.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/8cfedf396cd2e86092c03fa0dcb1fb3c978965.png[/img]
2010 Contests, 3
Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.