Found problems: 25757
1997 All-Russian Olympiad Regional Round, 8.3
On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.
2014 Sharygin Geometry Olympiad, 9
Two circles $\omega_1$ and $\omega_2$ touching externally at point $L$ are inscribed into angle $BAC$. Circle $\omega_1$ touches ray $AB$ at point $E$, and circle $\omega_2$ touches ray $AC$ at point $M$. Line $EL$ meets $\omega_2$ for the second time at point $Q$. Prove that $MQ\parallel AL$.
2010 Peru MO (ONEM), 4
A parallelepiped is said to be [i]integer [/i] when at least one of its edges measures a integer number of units. We have a group of integer parallelepipeds with which a larger parallelepiped is assembled, which has no holes inside or on its edge. Prove that the assembled parallelepiped is also integer.
Example. The following figure shows an assembled parallelepiped with a certain group of integer parallelepipeds.
[img]https://cdn.artofproblemsolving.com/attachments/3/7/f88954d6fe3a59fd2db6dcee9dddb120012826.png[/img]
2019 Novosibirsk Oral Olympiad in Geometry, 3
The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square.
(Distance in the picture is measured horizontally from the midpoint of the side of the square.)
[img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]
2023 Oral Moscow Geometry Olympiad, 3
Given is a triangle $ABC$ and $M$ is the midpoint of the minor arc $BC$. Let $M_1$ be the reflection of $M$ with respect to side $BC$. Prove that the nine-point circle bisects $AM_1$.
1963 German National Olympiad, 6
Consider a pyramid $ABCD$ whose base $ABC$ is a triangle. Through a point $M$ of the edge $DA$, the lines $MN$ and $MP$ on the plane of the surfaces $DAB$ and $DAC$ are drawn respectively, such that $N$ is on $DB$ and $P$ is on $DC$ and $ABNM$ , $ACPM$ are cyclic quadrilaterals.
a) Prove that $BCPN$ is also a cyclic quadrilateral.
b) Prove that the points $A,B,C,M,N, P$ lie on a sphere.
2010 Nordic, 2
Three circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ share a common point of intersection $O$. The other common point of $\Gamma_A$ and $\Gamma_B$ is $C$, that of $\Gamma_A$ and $\Gamma_C$ is $B$, and that of $\Gamma_C$ and $\Gamma_B$ is $A$. The line $AO$ intersects the circle $\Gamma_A$ in the point $X \ne O$. Similarly, the line $BO$ intersects the circle $\Gamma_B$ in the point $Y \ne O$, and the line $CO$ intersects the circle $\Gamma_C$ in the point $Z \ne O$. Show that
\[\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.\]
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 $BC$.
2019 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle in which $AB < AC, D$ is the foot of the altitude from $A, H$ is the orthocenter, $O$ is the circumcenter, $M$ is the midpoint of the side $BC, A'$ is the reflection of $A$ across $O$, and $S$ is the intersection of the tangents at $B$ and $C$ to the circumcircle. The tangent at $A'$ to the circumcircle intersects $SC$ and $SB$ at $X$ and $Y$ , respectively. If $M,S,X,Y$ are concyclic, prove that lines $OD$ and $SA'$ are parallel.
1996 Polish MO Finals, 2
Let $P$ be a point inside a triangle $ABC$ such that $\angle PBC = \angle PCA < \angle PAB$. The line $PB$ meets the circumcircle of triangle $ABC$ at a point $E$ (apart from $B$). The line $CE$ meets the circumcircle of triangle $APE$ at a point $F$ (apart from $E$). Show that the ratio $\frac{\left|APEF\right|}{\left|ABP\right|}$ does not depend on the point $P$, where the notation $\left|P_1P_2...P_n\right|$ stands for the area of an arbitrary polygon $P_1P_2...P_n$.
2019 ELMO Problems, 4
Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order.
[i]Proposed by Vincent Huang[/i]
2012 Flanders Math Olympiad, 4
In $\vartriangle ABC, \angle A = 66^o$ and $| AB | <| AC |$. The outer bisector in $A$ intersects $BC$ in $D$ and $| BD | = | AB | + | AC |$. Determine the angles of $\vartriangle ABC$.
2019 Regional Competition For Advanced Students, 2
The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.
2019 Swedish Mathematical Competition, 2
Segment $AB$ is the diameter of a circle. Points $C$ and $D$ lie on the circle. The rays $AC$ and $AD$ intersect the tangent to the circle at point $B$ at points $P$ and $Q$, respectively. Show that points $C, D, P$ and $Q$ lie on a circle.
1979 Romania Team Selection Tests, 1.
Let $\triangle ABC$ be a triangle with $\angle BAC=60^\circ$, $M$ be a point in its interior and $A',\, B',\, C'$ be the orthogonal projections of $M$ on the sides $BC,\, CA,\, AB$. Determine the locus of $M$ when the sum $A'B+B'C+C'A$ is constant.
[i]Horea Călin Pop[/i]
2020 March Advanced Contest, 2
An acute triangle \(ABC\) has circumcircle \(\Gamma\) and circumcentre \(O\). The incentres of \(AOB\) and \(AOC\) are \(I_b\) and \(I_c\) respectively. Let \(M\) be the the point on \(\Gamma\) such that \(MB = MC\) and \(M\) lies on the same side of \(BC\) as \(A\). Prove that the points \(M\), \(A\), \(I_b\), and \(I_c\) are concyclic.
2003 China Second Round Olympiad, 1
From point $P$ outside a circle draw two tangents to the circle touching at $A$ and $B$. Draw a secant line intersecting the circle at points $C$ and $D$, with $C$ between $P$ and $D$. Choose point $Q$ on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.
JBMO Geometry Collection, 1999
Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$).
[i]Greece[/i]
2003 Purple Comet Problems, 4
The lengths of the diagonals of a rhombus are, in inches, two consecutive integers. The area of the rhombus is $210$ sq. in. Find its perimeter, in inches.
2023 Olympic Revenge, 5
Let $ABCD$ be a circumscribed quadrilateral and $T=AC\cap BD$. Let $I_1$, $I_2$, $I_3$, $I_4$ the incenters of $\Delta TAB$, $\Delta TBC$, $TCD$, $TDA$, respectively, and $J_1$, $J_2$, $J_3$, $J_4$ the incenters of $\Delta ABC$, $\Delta BCD$, $\Delta CDA$, $\Delta DAB$. Show that $I_1I_2I_3I_4$ is a cyclic quadrilateral and its center is $J_1J_3\cap J_2J_4$
2017 Saint Petersburg Mathematical Olympiad, 2
Given a triangle $ABC$, there’s a point $X$ on the side $AB$ such that $2BX = BA + BC$. Let $Y$ be the point symmetric to the incenter $I$ of triangle $ABC$, with respect to point $X$. Prove that $YI_B\perp AB$ where $I_B$ is the $B$-excenter of triangle $ABC$.
2014 Contests, 1
Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.
1986 Federal Competition For Advanced Students, P2, 1
Show that a square can be inscribed in any regular polygon.
2012 Dutch IMO TST, 5
Let $\Gamma$ be the circumcircle of the acute triangle $ABC$. The angle bisector of angle $ABC$ intersects $AC$ in the point $B_1$ and the short arc $AC$ of $\Gamma$ in the point $P$. The line through $B_1$ perpendicular to $BC$ intersects the short arc $BC$ of $\Gamma$ in $K$. The line through $B$ perpendicular to $AK$ intersects $AC$ in $L$. Prove that $K, L$ and $P$ lie on a line.
1969 IMO Longlists, 2
$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$
$(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$
$(c)$ Find the locus of the centers of these hyperbolas.
$(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$