Found problems: 25757
2014 ELMO Shortlist, 2
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent.
[i]Proposed by Yang Liu[/i]
1977 Swedish Mathematical Competition, 2
There is a point inside an equilateral triangle side $d$ whose distance from the vertices is $3, 4, 5$. Find $d$.
2015 Costa Rica - Final Round, G4
Consider $\vartriangle ABC$, right at $B$, let $I$ be its incenter and $F,D,E$ the points where the circle inscribed on sides AB, $BC$ and $AC$, respectively. If $M$ is the intersection point of $CI$ and $EF$, and $N$ is the intersection point of $DM$ and $AB$. Prove that $AN = ID$.
2014 National Olympiad First Round, 9
Let $D$ be a point on side $[BC]$ of $\triangle ABC$ such that $|AB|=3, |CD|=1$ and $|AC|=|BD|=\sqrt{5}$. If the $B$-altitude of $\triangle ABC$ meets $AD$ at $E$, what is $|CE|$?
$
\textbf{(A)}\ \dfrac{2}{\sqrt{5}}
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ \dfrac{2}{\sqrt{3}}
\qquad\textbf{(D)}\ \dfrac{\sqrt{5}}{2}
\qquad\textbf{(E)}\ \dfrac{3}{2}
$
1982 IMO Longlists, 54
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.
1998 Romania National Olympiad, 3
In the exterior of the triangle $ABC$ with $m(\angle B) > 45^o$, $m(\angle C) >45°^o$ one constructs the right isosceles triangles $ACM$ and $ABN$ such that $m(\angle CAM) = m(\angle BAN) = 90^o$ and, in the interior of $ABC$, the right isosceles triangle $BCP$, with $m(\angle P) = 90^o$. Show that the triangle $MNP$ is a right isosceles triangle.
2012 Sharygin Geometry Olympiad, 6
Let $ABC$ be an isosceles triangle with $BC = a$ and $AB = AC = b$. Segment $AC$ is the base of an isosceles triangle $ADC$ with $AD = DC = a$ such that points $D$ and $B$ share the opposite sides of AC. Let $CM$ and $CN$ be the bisectors in triangles $ABC$ and $ADC$ respectively. Determine the circumradius of triangle $CMN$.
(M.Rozhkova)
Kyiv City MO Seniors Round2 2010+ geometry, 2017.11.2
The median $CM$ is drawn in the triangle $ABC$ intersecting bisector angle $BL$ at point $O$. Ray $AO$ intersects side $BC$ at point $K$, beyond point $K$ draw the segment $KT = KC$. On the ray $BC$ beyond point $C$ draw a segment $CN = BK$. Prove that is a quadrilateral $ABTN$ is cyclic if and only if $AB = AK$.
(Vladislav Yurashev)
1990 USAMO, 5
An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $\, M \,$ and $\, N \,$, and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q \,$. Prove that the points $\, M, N, P, Q \,$ lie on a common circle.
2015 Swedish Mathematical Competition, 1
Given the acute triangle $ABC$. A diameter of the circumscribed circle of the triangle intersects the sides $AC$ and $BC$, dividing the side $BC$ in half. Show that the same diameter divides the side $AC$ in a ratio of $1: 3$, calculated from $A$, if and only if $\tan B = 2 \tan C$.
2004 Italy TST, 1
Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$.
$(a)$ Prove that $t$ is parallel to $AC$.
$(b)$ Prove that the lines $r,s,t$ are concurrent.
2018 PUMaC Geometry B, 2
Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.
1984 Bundeswettbewerb Mathematik, 2
Given is a regular $n$-gon with circumradius $1$. $L$ is the set of (different) lengths of all connecting segments of its endpoints. What is the sum of the squares of the elements of $L$?
2001 German National Olympiad, 6 (12)
Let $ABC$ be a triangle with $\angle A = 90^o$ and $\angle B < \angle C$. The tangent at $A$ to the circumcircle $k$ of $\vartriangle ABC$ intersects line $BC$ at $D$. Let $E$ be the reflection of $A$ in $BC$. Also, let $X$ be the feet of the perpendicular from $A$ to $BE$ and let $Y$ be the midpoint of $AX$. Line $BY$ meets $k$ again at $Z$. Prove that line $BD$ is tangent to the circumcircle of $\vartriangle ADZ$.
2012 Math Prize for Girls Olympiad, 1
Let $A_1A_2 \dots A_n$ be a polygon (not necessarily regular) with $n$ sides. Suppose there is a translation that maps each point $A_i$ to a point $B_i$ in the same plane. For convenience, define $A_0 = A_n$ and $B_0 = B_n$. Prove that
\[
\sum_{i=1}^{n} (A_{i-1} B_{i})^2 = \sum_{i=1}^{n} (B_{i-1} A_{i})^2 \, .
\]
1987 AMC 12/AHSME, 14
$ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Then $\sin \theta=$
[asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((0,0)--(2,1));
draw((0,0)--(1,2));
label("A", (0,0), SW);
label("B", (0,2), NW);
label("C", (2,2), NE);
label("D", (2,0), SE);
label("M", (1,2), N);
label("N", (2,1), E);
label("$\theta$", (.5,.5), SW);
[/asy]
$ \textbf{(A)}\ \frac{\sqrt{5}}{5} \qquad\textbf{(B)}\ \frac{3}{5} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{5} \qquad\textbf{(D)}\ \frac{4}{5} \qquad\textbf{(E)}\ \text{none of these} $
2005 National High School Mathematics League, 15
$A(1,1)$ is a point on parabola $y=x^2$. Draw the tangent line of the parabola that passes $A$, the line intersects $x$-axis at $D$, intersects $y$-axis at $B$. $C$ is a point on the parabola, and $E$ is a point on segment $AC$, such that $\frac{AE}{EC}=\lambda_1$, $F$ is a point on segment $BC$, such that $\frac{BF}{FC}=\lambda_2$. If $\lambda_1+\lambda_2=1$, $CD$ and $EF$ intersect at $P$. When $C$ moves, find the path equation of $P$.
2005 Junior Balkan Team Selection Tests - Romania, 9
Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that \[ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \]
[i]Dinu Serbanescu[/i]
2015 Junior Balkan Team Selection Tests - Romania, 1
Let $ABC$ be an acute triangle with $AB \neq AC$ . Also let $M$ be the midpoint of the side $BC$ , $H$ the orthocenter of the triangle $ABC$ , $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$ . Prove that $O_1AMO_2$ is a parallelogram .
1978 Austrian-Polish Competition, 2
A parallelogram is inscribed into a regular hexagon so that the centers of symmetry of both figures coincide. Prove that the area of the parallelogram does not exceed $2/3$ the area of the hexagon.
1992 Irish Math Olympiad, 4
A convex pentagon has the property that each of its diagonals cuts off a triangle of unit area. Find the area of the pentagon.
2003 Hong kong National Olympiad, 3
Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.
The Golden Digits 2024, P2
We are given an infinite set of points in the plane such that any two of them have a distance of at most one. Prove that all the axes of symmetry of this set are concurrent, provided that there are at least two of them.
[i]Proposed by David Anghel[/i]
2014 BMT Spring, 9
Let $ABC$ be a triangle. Construct points $B'$ and $ C'$ such that $ACB'$ and $ABC'$ are equilateral triangles that have no overlap with $ \vartriangle ABC$. Let $BB'$ and $CC'$ intersect at X. If $AX = 3$, $BC = 4$, and $CX = 5$, find the area of quadrilateral $BCB'C'$.
.
2018 Iranian Geometry Olympiad, 3
Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.
Proposed by Mahdi Etesamifard