Found problems: 25757
2022 Greece National Olympiad, 1
Let $ABC$ be a triangle such that $AB<AC<BC$. Let $D,E$ be points on the segment $BC$ such that $BD=BA$ and $CE=CA$. If $K$ is the circumcenter of triangle $ADE$, $F$ is the intersection of lines $AD,KC$ and $G$ is the intersection of lines $AE,KB$, then prove that the circumcircle of triangle $KDE$ (let it be $c_1$), the circle with center the point $F$ and radius $FE$ (let it be $c_2$) and the circle with center $G$ and radius $GD$ (let it be $c_3$) concur on a point which lies on the line $AK$.
2013 Bosnia Herzegovina Team Selection Test, 1
Triangle $ABC$ is right angled at $C$. Lines $AM$ and $BN$ are internal angle bisectors.
$AM$ and $BN$ intersect altitude $CH$ at points $P$ and $Q$ respectively.
Prove that the line which passes through the midpoints of segments $QN$ and $PM$ is parallel to $AB$.
2009 Sharygin Geometry Olympiad, 17
Given triangle $ ABC$ and two points $ X$, $ Y$ not lying on its circumcircle. Let $ A_1$, $ B_1$, $ C_1$ be the projections of $ X$ to $ BC$, $ CA$, $ AB$, and $ A_2$, $ B_2$, $ C_2$ be the projections of $ Y$. Prove that the perpendiculars from $ A_1$, $ B_1$, $ C_1$ to $ B_2C_2$, $ C_2A_2$, $ A_2B_2$, respectively, concur if and only if line $ XY$ passes through the circumcenter of $ ABC$.
2005 Danube Mathematical Olympiad, 3
Let $\mathcal{C}$ be a circle with center $O$, and let $A$ be a point outside the circle. Let the two tangents from the point $A$ to the circle $\mathcal{C}$ meet this circle at the points $S$ and $T$, respectively. Given a point $M$ on the circle $\mathcal{C}$ which is different from the points $S$ and $T$, let the line $MA$ meet the perpendicular from the point $S$ to the line $MO$ at $P$.
Prove that the reflection of the point $S$ in the point $P$ lies on the line $MT$.
2023 BMT, 8
Circle $\omega_1$ is centered at $O_1$ with radius $3$, and circle $\omega_2$ is centered at $O_2$ with radius $2$. Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $X$, $Z$, respectively, and intersects segment $\overline{O_1O_2}$ at $Y$ . The circle through $O_1$, $X$, $Y$ has center $O_3$, and the circle through $O_2$, $Y$ , $Z$ has center $O_4$. Given that $O_1O_2 = 13$, find $O_3O_4$.
2015 Indonesia MO Shortlist, G4
Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.
2005 International Zhautykov Olympiad, 3
Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! \equal{} S)$ in the space satisfing the equation $ |cos ASD \minus{} 2cosBSD \minus{} 2 cos CSD| \equal{} 3$.
2009 Belarus Team Selection Test, 1
Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$, $K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$.
I. Voronovich
2015 Iran Geometry Olympiad, 2
Let $ABC$ be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.
by Mahdi Etesami Fard
2021 Princeton University Math Competition, 9
Let $AX$ be a diameter of a circle $\Omega$ with radius $10$, and suppose that $C$ lies on $\Omega$ so that $AC = 16$. Let $D$ be the other point on $\Omega$ so $CX = CD$. From here, define $D'$ to be the reflection of $D$ across the midpoint of $AC$, and $X'$ to be the reflection of $X$ across the midpoint of $CD$. If the area of triangle $CD'X'$ can be written as $\frac{p}{q}$ , where $p, q$ are relatively prime, find $p + q$.
2006 Harvard-MIT Mathematics Tournament, 1
Octagon $ABCDEFGH$ is equiangular. Given that $AB=1$, $BC=2$, $CD=3$, $DE=4$, and $EF=FG=2$, compute the perimeter of the octagon.
2018 HMNT, 1
Square $CASH$ and regular pentagon $MONEY$ are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?
2012 Today's Calculation Of Integral, 798
Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$.
(1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin.
(2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized.
50 points
2019 CMIMC, 2
Suppose $X, Y, Z$ are collinear points in that order such that $XY = 1$ and $YZ = 3$. Let $W$ be a point such that $YW = 5$, and define $O_1$ and $O_2$ as the circumcenters of triangles $\triangle WXY$ and $\triangle WYZ$, respectively. What is the minimum possible length of segment $\overline{O_1O_2}$?
2001 Kazakhstan National Olympiad, 2
In the acute triangle $ ABC $, $ L $, $ H $ and $ M $ are the intersection points of bisectors, altitudes and medians, respectively, and $ O $ is the center of the circumscribed circle. Denote by $ X $, $ Y $ and $ Z $ the intersection points of $ AL $, $ BL $ and $ CL $ with a circle, respectively. Let $ N $ be a point on the line $ OL $ such that the lines $ MN $ and $ HL $ are parallel. Prove that $ N $ is the intersection point of the medians of $ XYZ $.
2012 Tournament of Towns, 5
Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.
2021 Saint Petersburg Mathematical Olympiad, 5
Given is an isosceles trapezoid $ABCD$, such that $AD$ and $BC$ are bases and $AD=2AB$, and it is inscribed in a circle $c$. Points $E$ and $F$ are selected on a circle $c$ so that $AC$ || $DE$ and $BD$ || $AF$. The line $BE$ intersects lines $AC$ and $AF$ at points $X$ and $Y$, respectively. Prove that the circumcircles of triangles $BCX$ and $EFY$ are tangent to each other.
2014 USAMTS Problems, 4:
A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.
2003 Tournament Of Towns, 3
Point $M$ is chosen in triangle $ABC$ so that the radii of the circumcircles of triangles $AMC, BMC$, and $BMA$ are no smaller than the radius of the circumcircle of $ABC$. Prove that all four radii are equal.
2025 China Team Selection Test, 21
Given a circle \( \omega \) and two points \( A \) and \( B \) outside \( \omega \), a quadrilateral \( PQRS \) is defined as[i] "good"[/i] if \( P, Q, R, S \) are four distinct points on \( \omega \) in order, and lines \( PQ \) and \( RS \) intersect at \( A \) and lines \( PS \) and \( QR \) intersect at \( B \).
For a quadrilateral \( T \), let \( S_T \) denote its area. If there exists a [i]good[/i] quadrilateral, prove that there exists [i]good[/i] quadrilateral \( T \) such that for any good quadrilateral $T_1 (T_1 \neq T)$, \( S_{T_1} < S_T \).
2001 Kazakhstan National Olympiad, 7
Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.
1987 IMO Longlists, 40
The perpendicular line issued from the center of the circumcircle to the bisector of angle $C$ in a triangle $ABC$ divides the segment of the bisector inside $ABC$ into two segments with ratio of lengths $\lambda$. Given $b = AC$ and $a = BC$, find the length of side $c.$
2014 Denmark MO - Mohr Contest, 3
The points $C$ and $D$ lie on a halfline from the midpoint $M$ of a segment $AB$, so that $|AC| = |BD|$. Prove that the angles $u = \angle ACM$ and $v = \angle BDM$ are equal.
[img]https://1.bp.blogspot.com/-tQEJ1VBCa8U/XzT7IhwlZHI/AAAAAAAAMVI/xpRdlj5Rl64VUt_tCRsQ1UxIsv_SGrMlACLcBGAsYHQ/s0/2014%2BMohr%2Bp3.png[/img]
2010 ELMO Shortlist, 5
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three.
[i]Carl Lian.[/i]
2009 Romania National Olympiad, 1
On the sides $ AB,AC $ of a triangle $ ABC, $ consider the points $ M, $ respectively, $ N $ such that $ M\neq A\neq N $ and $ \frac{MB}{MA}\neq\frac{NC}{NA}. $ Show that the line $ MN $ passes through a point not dependent on $ M $ and $ N. $