Found problems: 25757
2012 IMO Shortlist, G6
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$. The points $D,E$ and $F$ on the sides $BC,CA$ and $AB$ respectively are such that $BD+BF=CA$ and $CD+CE=AB$. The circumcircles of the triangles $BFD$ and $CDE$ intersect at $P \neq D$. Prove that $OP=OI$.
2000 Moldova National Olympiad, Problem 7
In an isosceles triangle $ABC$ with $BC=AC$, $I$ is the incenter and $O$ the circumcenter. The line through $I$ parallel to $AC$ meets $BC$ at $D$. Prove that the lines $DO$ and $BI$ are perpendicular.
2017 District Olympiad, 2
Let $ ABC $ be a triangle in which $ O,I, $ are the circumcenter, respectively, incenter. The mediators of $ IA,IB,IC, $ form a triangle $ A_1B_1C_1. $ Show that $ \overrightarrow{OI}=\overrightarrow{OA_1} +\overrightarrow{OA_2} +\overrightarrow{OA_3} . $
2016 Singapore Junior Math Olympiad, 3
In the triangle $ABC$, $\angle A=90^\circ$, the bisector of $\angle B$ meets the altitude $AD$ at the point $E$, and the bisector of $\angle CAD$ meets the side $CD$ at $F$. The line through $F$ perpendicular to $BC$ intersects $AC$ at $G$. Prove that $B,E,G$ are collinear.
2017 Sharygin Geometry Olympiad, 1
Let $ABCD$ be a cyclic quadrilateral with $AB=BC$ and $AD = CD$. A point $M$ lies on the minor arc $CD$ of its circumcircle. The lines $BM$ and $CD$ meet at point $P$, the lines $AM$ and $BD$ meet at point $Q$. Prove that $PQ \parallel AC$.
1952 Polish MO Finals, 4
Prove that if the angles $ A $, $ B $, $ C $ of a triangle satisfy the equation
$$\cos 3A + \cos 3B + \cos 3C = 1,$$
then one of these angles equals $120^\circ $.
2018 Moldova EGMO TST, 7
Let $ABCD$ be a isosceles trapezoid with $AB \| CD $ , $AD=BC$, $ AC \cap BD = $ { $O$ }. $ M $ is the midpoint of the side $AD$ . The circumcircle of triangle $ BCM $ intersects again the side $AD$ in $K$. Prove that $OK \| AB $ .
2010 Math Prize For Girls Problems, 19
Let $S$ be the set of 81 points $(x, y)$ such that $x$ and $y$ are integers from $-4$ through $4$. Let $A$, $B$, and $C$ be random points chosen independently from $S$, with each of the 81 points being equally likely. (The points $A$, $B$, and $C$ do not have to be different.) Let $K$ be the area of the (possibly degenerate) triangle $ABC$. What is the expected value (average value) of $K^2$ ?
2007 Harvard-MIT Mathematics Tournament, 9
$\triangle ABC$ is right angled at $A$. $D$ is a point on $AB$ such that $CD=1$. $AE$ is the altitude from $A$ to $BC$. If $BD=BE=1$, what is the length of $AD$?
2007 Indonesia TST, 1
Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha\equal{}\angle BPC\minus{}\angle BAC, \quad \beta\equal{}\angle CPA\minus{}\angle \angle CBA, \quad \gamma\equal{}\angle APB\minus{}\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]
2020 Moldova Team Selection Test, 8
In $\Delta ABC$ the angles $ABC$ and $ACB$ are acute. Let $M$ be the midpoint of $AB$. Point $D$ is on the half-line $(CB$ such that $ B \in (CD)$ and $\angle DAB= \angle BCM$. Perpendicular from $B$ to line $CD$ intersects the line bisector of $AB$ in $E$. Prove that $DE$ and $AC$ are perpendicular.
2012 JHMT, 7
What is the radius of the largest sphere that fits inside an octahedron of side length $1$?
2016 Singapore Senior Math Olympiad, 4
Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90^o$ or $270^o$. If the lengths of its sides are odd integers, prove that its area is an even integer.
2019 Moroccan TST, 1
Let $ABCD$ a cyclique quadrilateral. We consider the Following points: $A'$ the orthogonal projection of $A$ over $(BD)$, $B'$ the orthogonal projection of $B$ over $(AC)$, $C'$ the orthgonal projection of $C$ over $(BD)$ and $D'$ being the orthogonal projection of $D$ over $(AC)$
Prove that $A', B', C'$ and $D'$.
2025 Francophone Mathematical Olympiad, 3
Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.
2001 Moldova National Olympiad, Problem 3
For an arbitrary point $D$ on side $BC$ of an acute-angled triangle $ABC$, let $O_1$ and $O_2$ be the circumcenters of the triangles $ABD$ and $ACD$, and $O$ be the circumcenter of the triangle $AO_1O_2$. Find the locus of $O$ when $D$ moves across $BC$.
2014 BMT Spring, 10
A plane intersects a sphere of radius $10$ such that the distance from the center of the sphere to the plane is $9$. The plane moves toward the center of the bubble at such a rate that the increase in the area of the intersection of the plane and sphere is constant, and it stops once it reaches the center of the circle. Determine the distance from the center of the sphere to the plane after two-thirds of the time has passed.
2010 AMC 10, 14
Triangle $ ABC$ has $ AB \equal{} 2 \cdot AC$. Let $ D$ and $ E$ be on $ \overline{AB}$ and $ \overline{BC}$, respectively, such that $ \angle{BAE} \equal{} \angle{ACD}.$ Let $ F$ be the intersection of segments $ AE$ and $ CD$, and suppose that $ \triangle{CFE}$ is equilateral. What is $ \angle{ACB}$?
$ \textbf{(A)}\ 60^{\circ}\qquad \textbf{(B)}\ 75^{\circ}\qquad \textbf{(C)}\ 90^{\circ}\qquad \textbf{(D)}\ 105^{\circ}\qquad \textbf{(E)}\ 120^{\circ}$
1988 Tournament Of Towns, (189) 2
A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle MAC = \angle MCD = x$ . Find $\angle ABM$.
2024 Pan-African, 2
In triangle $ABC$,let $M$ be the midpoint of the side $BC$,and $N$ is the midpoint of the segment $AM$,the circle going through $N$ and tangent the line $AC$ at $A$ intersects the segment $AB$ again in $P$.
prove that the circumcircle of triangle $BPM$ is tangent the line $AM$
1985 Czech And Slovak Olympiad IIIA, 1
A regular $1985$-gon is given in the plane. Let's pass a straight line through each side of it. Determine the number of parts into which these lines divide the plane.
2001 AMC 10, 18
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
$ \textbf{(A)} \ 50 \qquad \textbf{(B)} \ 52 \qquad \textbf{(C)} \ 54 \qquad \textbf{(D)} \ 56 \qquad \textbf{(E)} \ 58 \qquad$
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path p2=(0,1)--(1,1)--(1,0);
path p3=(2,0)--(2,1)--(3,1);
path p4=(3,2)--(2,2)--(2,3);
path p5=(1,3)--(1,2)--(0,2);
path p6=(1,1)--(2,2);
path p7=(2,1)--(1,2);
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for(int i=0; i<3; ++i)
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draw(shift(3*i,3*j)*p);
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1959 Miklós Schweitzer, 2
[b]2.[/b] Omit the vertices of a closed rectangle; the configuration obtained in such a way will be called a reduced rectangle. Prove tha the set-union of any system of reduced rectangles with parallel sides is equal to the union of countably many elements of the system. [b](St. 3)[/b]
2020 Jozsef Wildt International Math Competition, W9
In any triangle $ABC$ prove that the following relationship holds:
$$\begin{vmatrix}(b+c)^2&a^2&a^2\\b^2&(c+a)^2&b^2\\c^2&c^2&(a+b)^2\end{vmatrix}\ge93312r^6$$
[i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]
2023 CUBRMC, 2
The concave decagon shown below is embedded in the Cartesian coordinate plane such that all of its vertices have integer coordinates. Two opposite edges have length $5$, whereas the remaining eight edges have length $\sqrt{10}$. Every pair of opposite edges is parallel. The sides of the decagon do not intersect each other, and the decagon has vertical and horizontal axes of symmetry. Find the area of the decagon.
[img]https://cdn.artofproblemsolving.com/attachments/1/5/daa4ab3d71af4b3274cd222f9a091eea3be705.png[/img]