Found problems: 25757
May Olympiad L2 - geometry, 2000.2
Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon
2010 Dutch BxMO TST, 4
The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.
2014 South East Mathematical Olympiad, 6
Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.
2017 Iran Team Selection Test, 6
In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ , CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$
[i] Proposed by Iman Maghsoudi[/i]
1994 All-Russian Olympiad Regional Round, 9.6
Point $ P$ is taken inside a right angle $ KLM$. A circle $ S_1$ with center $ O_1$ is tangent to the rays $ LK,LP$ of angle $ KLP$ at $ A,D$ respectively. A circle $ S_2$ with center $ O_2$ is tangent to the rays of angle $ MLP$, touching $ LP$ at $ B$. Suppose $ A,B,O_1$ are collinear. Let $ O_2D,KL$ meet at $ C$. Prove that $ BC$ bisects angle $ ABD$.
2023 Quang Nam Province Math Contest (Grade 11), Problem 5
a) Given an acute triangle $ABC(AB>AC).$ The circle $(O)$ with diameter $BC$ intersects $AB,AC$ at $F,E$, respectively. Let $H$ be the intersection point of $BE,CF,$ the line $AH$ intersects the line $BC$ at $D,$ the line $EF$ intersects the line $BC$ at $K.$ The line passing through $D$ and parallel to $EF$ intersects $AB,AC$ at $M,N,$ respectively.
Prove that: $M,O,N,K$ are on the same circle.
b) Given $\triangle ABC, \angle BAC=\angle BCA=30^{\circ}.$ $D,E,F$ are moving points on the side $AB,BC,CA$ such that: $\angle BFD=\angle BFE=60^{\circ}.$ Let $p,p_1$ be the perimeter of $\triangle ABC,\triangle DEF,$ respectively. Prove that: $p\le 2p_1.$
2008 Indonesia MO, 3
Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to
$ \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}}$
(hmm,, looks familiar, isn't it? :wink: )
1987 AMC 12/AHSME, 18
It takes $A$ algebra books (all the same thickness) and $H$ geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, $S$ of the algebra books and $M$ of the geometry books would fill the same shelf. Finally, $E$ of the algebra books alone would fill this shelf. Given that $A, H, S, M, E$ are distinct positive integers, it follows that $E$ is
$ \textbf{(A)}\ \frac{AM+SH}{M+H} \qquad\textbf{(B)}\ \frac{AM^2+SH^2}{M^2+H^2} \qquad\textbf{(C)}\ \frac{AH-SM}{M-H} \qquad\textbf{(D)}\ \frac{AM-SH}{M-H} \qquad\textbf{(E)}\ \frac{AM^2-SH^2}{M^2-H^2} $
2007 All-Russian Olympiad Regional Round, 10.4
Given a triangle $ ABC$. A circle passes through vertices $ B$ and $ C$ and intersects sides $ AB$ and $ AC$ at points $ D$ and $ E$, respectively. Segments $ CD$ and $ BE$ intersect at point $ O$. Denote the incenters of triangles $ ADE$ and $ ODE$ by $ M$ and $ N$, respectiely. Prove that the midpoint of the smaller arc $ DE$ lies on line $ MN$.
2006 MOP Homework, 7
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $T$. Let $X$ be a point on circle $\omega_1$. Line $l_1$ is tangent to circle $\omega_1$ and $X$, and line $l$ intersects circle $\omega_2$ at $A$ and $B$. Line $XT$ meets circle $\omega$ at $S$. Point $C$ lies on arc $TS$ (of circle $\omega_2$, not containing points $A$ and $B$). Point $Y$ lies on circle $\omega_1$ and line $YC$ is tangent to circle $\omega_1$. Let $I$ be the intersection of lines $XY$ ad $SC$. Prove that...
a) points $C$, $T$, $Y$, $I$ lie on a circle
(B) $I$ is an excenter of triangle $ABC$.
2014 Serbia JBMO TST, 3
Consider parallelogram $ABCD$, with acute angle at $A$, $AC$ and $BD$ intersect at $E$. Circumscribed circle of triangle $ACD$ intersects $AB$, $BC$ and $BD$ at $K$, $L$ and $P$ (in that order). Then, circumscribed circle of triangle $CEL$ intersects $BD$ at $M$. Prove that: $$KD*KM=KL*PC$$
2009 Sharygin Geometry Olympiad, 10
Let $ ABC$ be an acute triangle, $ CC_1$ its bisector, $ O$ its circumcenter. The perpendicular from $ C$ to $ AB$ meets line $ OC_1$ in a point lying on the circumcircle of $ AOB$. Determine angle $ C$.
2007 Miklós Schweitzer, 9
Let $A$ and $B$ be two triangles on the plane such that the interior of both contains the origin and for each circle $C_r$ centered at the origin $|C_r \cap A|=|C_r\cap B|$ (where $|\cdot |$ is the arc-length measure). Prove that $A$ and $B$ are congruent. Does this statement remain true if the origin is on the border of $A$ or $B$?
(translated by Miklós Maróti)
2014 Sharygin Geometry Olympiad, 2
In a quadrilateral $ABCD$ angles $A$ and $C$ are right. Two circles with diameters $AB$ and $CD$ meet at points $X$ and $Y$ . Prove that line $XY$ passes through the midpoint of $AC$.
(F. Nilov )
2025 PErA, P3
Let \( ABC \) be an equilateral triangle with circumcenter \( O \). Let \( X \) and \( Y \) be two points on segments \( AB \) and \( AC \), respectively, such that \( \angle XOY = 60^\circ \). If \( T \) is the reflection of \( O \) with respect to line \( XY \), prove that lines \( BT \) and \( OY \) are parallel.
2009 AMC 10, 6
A circle of radius $ 2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
filldraw(Arc((0,0),4,0,180)--cycle,gray,black);
filldraw(Circle((0,2),2),white,black);
dot((0,2));
draw((0,2)--((0,2)+2*dir(60)));
label("$2$",midpoint((0,2)--((0,2)+2*dir(60))),SE);[/asy]$ \textbf{(A)}\ \frac{1}{2}\qquad
\textbf{(B)}\ \frac{\pi}{6}\qquad
\textbf{(C)}\ \frac{2}{\pi}\qquad
\textbf{(D)}\ \frac{2}{3}\qquad
\textbf{(E)}\ \frac{3}{\pi}$
2020 HMNT (HMMO), 2
Let $T$ be a trapezoid with two right angles and side lengths $4, 4, 5,$ and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into $4$ regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240d$.
2023 BMT, 6
Let rectangle $ABCD$ have side lengths $AB = 8$, $BC = 6$. Let $ABCD$ be inscribed in a circle with center $O$, as shown in the diagram. Let $M$ be the midpoint of side $\overline{AB}$, and let $X$ be the intersection of ray $\overrightarrow{MO}$ with the circle. Compute the length $AX$.
[img]https://cdn.artofproblemsolving.com/attachments/6/0/a13e7ec6798f57d896265f61fa42df4c6cab15.png[/img]
1971 IMO Longlists, 14
Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.
2010 Bosnia And Herzegovina - Regional Olympiad, 2
It is given acute triangle $ABC$ with orthocenter at point $H$. Prove that $$AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}$$ where $a$, $b$ and $c$ are sides of a triangle, and $h_a$, $h_b$ and $h_c$ altitudes of $ABC$
2011 AMC 12/AHSME, 24
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
$ \textbf{(A)}\ \sqrt{15} \qquad\textbf{(B)}\ \sqrt{21} \qquad\textbf{(C)}\ 2\sqrt{6} \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 2\sqrt{7} $
2025 Romanian Master of Mathematics, 5
Let triangle $ABC$ be an acute triangle with $AB<AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. Let $\Gamma$ be the circle $BOC$. The line $AO$ and the circle of radius $AO$ centered at $A$ cross $\Gamma$ at $A’$ and $F$, respectively. Prove that $\Gamma$ , the circle on diameter $AA’$ and circle $AFH$ are concurrent.
[i]Proposed by Romania, Radu-Andrew Lecoiu[/i]
2002 Croatia National Olympiad, Problem 2
Consider the cube with the vertices $A(1,1,1)$, $B(-1,1,1)$, $C(-1,-1,1)$, $D(1,-1,1)$ and $A',B',C',D'$ symmetric to $A,B,C,D$ respectively with respect to the origin $O$. Let $T$ be a point not on the circumsphere of the cube and let $OT=d$. Denote $\alpha=\angle ATA'$, $\beta=\angle BTB'$, $\gamma=\angle CTC'$, $\delta=\angle DTD'$. Prove that
$$\tan^2\alpha+\tan^2\beta+\tan^2\gamma+\tan^2\delta=\frac{32d^2}{\left(d^2-3\right)^2}.$$
2025 CMIMC Geometry, 6
Points $A, B, C, D, E,$ and $F$ lie on a sphere with center $O$ and radius $R$ such that $\overline{AB}, \overline{CD},$ and $\overline{EF}$ are pairwise perpendicular and all meet at a point $X$ inside the sphere. If $AX=1, CX=\sqrt{2}, EX=2,$ and $OX=\tfrac{\sqrt{2}}{2},$ compute the sum of all possible values of $R^2.$
1997 Greece National Olympiad, 1
Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$.