Found problems: 25757
1984 Tournament Of Towns, (054) O2
In the convex pentagon $ABCDE$, $AE = AD$, $AB = AC$, and angle $CAD$ equals the sum of angles $AEB$ and $ABE$. Prove that segment $CD$ is double the length of median $AM$ of triangle $ABE$.
2014 Singapore Junior Math Olympiad, 3
In the triangle $ABC$, the bisector of $\angle A$ intersects the bisection of $\angle B$ at the point $I, D$ is the foot of the perpendicular from $I$ onto $BC$. Prove that the bisector of $\angle BIC$ is perpendicular to the bisector $\angle AID$.
Durer Math Competition CD Finals - geometry, 2020.C4
Albrecht likes to draw hexagons with all sides having equal length. He calls an angle of such a hexagon [i]nice [/i] if it is exactly $120^o$. He writes the number of its nice angles inside each hexagon. How many different numbers could Albrecht write inside the hexagons? Show examples for as many values as possible and give a reasoning why others cannot appear.
[i]Albrecht can also draw concave hexagons[/i]
Oliforum Contest I 2008, 3
Let $ C_1,C_2$ and $ C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $ i,j$, we define $ (r_{ij},s_{ij})$ as the two internal tangent lines of $ (C_i,C_j)$. Let $ r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ ABCA'B'C'$.
Prove that $ AA',BB'$ and $ CC'$ are concurrent.
[img]https://cdn.artofproblemsolving.com/attachments/1/2/5ef098966fc9f48dd06239bc7ee803ce4701e2.png[/img]
2021 Benelux, 3
A cyclic quadrilateral $ABXC$ has circumcentre $O$. Let $D$ be a point on line $BX$ such that $AD = BD$. Let $E$ be a point on line $CX$ such that $AE = CE$. Prove that the circumcentre of triangle $\triangle DEX$ lies on the perpendicular bisector of $OA$.
2015 IFYM, Sozopol, 6
The points $A_1$,$B_1$,$C_1$ are middle points of the arcs $\widehat{BC}, \widehat{CA}, \widehat{AB}$ of the circumscribed circle of $\Delta ABC$, respectively. The points $I_a,I_b,I_c$ are the reflections in the middle points of $BC,CA,AB$ of the center $I$ of the inscribed circle in the triangle. Prove that $I_a A_1,I_b B_1$, and $I_c C_1$ are concurrent.
2006 Estonia National Olympiad, 4
Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the
circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.
2010 Sharygin Geometry Olympiad, 5
A point $E$ lies on the altitude $BD$ of triangle $ABC$, and $\angle AEC=90^\circ.$ Points $O_1$ and $O_2$ are the circumcenters of triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2.$ Prove that the points $L,E,F$ are collinear.
1985 ITAMO, 11
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?
2024 BMT, 6
Let triangle $\triangle{ABC}$ be acute. Point $D$ is the foot of the altitude of $\triangle{ABC}$ from $A$ to $\overline{BC},$ and $E$ is the foot of the altitude of $\triangle{ABC}$ from $B$ to $\overline{AC}.$ Let $F$ denote the point of intersection between $\overline{BE}$ and $\overline{AD},$ and let $G$ denote the point of intersection between $\overline{CF}$ and $\overline{DE}.$ The areas of triangles $\triangle{EFG}, \triangle{CDG},$ and $\triangle{CEG}$ are $1,4,$ and $3,$ respectively. Find the area of $\triangle{ABC}.$
1996 Moldova Team Selection Test, 7
Let $ABCDA_1B_1C_1D_1$ be a cube. On the sides $AB{}$ and $AD{}$ there are the points $M{}$ and $N{}$, respectively, such that $AM+AN=AB$. Show that the measure of the dihedral angle between the planes $(MA_1C)$ and $(NA_1C)$ doe not depend on the positions of $M{}$ and $N{}$. Find this measure.
1999 Mexico National Olympiad, 3
A point $P$ is given inside a triangle $ABC$. Let $D,E,F$ be the midpoints of $AP,BP,CP$, and let $L,M,N$ be the intersection points of $ BF$ and $CE, AF$ and $CD, AE$ and $BD$, respectively.
(a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$.
(b) Prove that $DL,EM$, and $FN$ are concurrent.
2011 Indonesia TST, 3
Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define
\[
p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.
\]
Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?
1988 IMO Longlists, 13
Let $T$ be a triangle with inscribed circle $C.$ A square with sides of length $a$ is circumscribed about the same circle $C.$ Show that the total length of the parts of the edge of the square interior to the triangle $T$ is at least $2 \cdot a.$
1990 India Regional Mathematical Olympiad, 5
$P$ is any point inside a triangle $ABC$. The perimeter of the triangle $AB + BC + Ca = 2s$. Prove that $s < AP +BP +CP < 2s$.
2001 All-Russian Olympiad Regional Round, 8.8
Prove that any triangle can be cut by at most into $3$ parts, from which an isosceles triangle is formed.
Estonia Open Junior - geometry, 1995.2.1
A rectangle, whose one sidelength is twice the other side, is inscribed inside a triangles with sides $3$ cm, $4$ cm and $5$ cm, such that the long sides lies entirely on the long side of the triangle. The other two remaining vertices of the rectangle lie respectively on the other two sides of the triangle. Find the lengths of the sides of this rectangle.
2007 Today's Calculation Of Integral, 245
A sextic funtion $ y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma \ (\alpha < \beta < \gamma ).$
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma .$
created by kunny
2004 Bundeswettbewerb Mathematik, 3
Given two circles $k_1$ and $k_2$ which intersect at two different points $A$ and $B$. The tangent to the circle $k_2$ at the point $A$ meets the circle $k_1$ again at the point $C_1$. The tangent to the circle $k_1$ at the point $A$ meets the circle $k_2$ again at the point $C_2$. Finally, let the line $C_1C_2$ meet the circle $k_1$ in a point $D$ different from $C_1$ and $B$.
Prove that the line $BD$ bisects the chord $AC_2$.
Indonesia MO Shortlist - geometry, g10
Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$
2013 Online Math Open Problems, 10
In convex quadrilateral $AEBC$, $\angle BEA = \angle CAE = 90^{\circ}$ and $AB = 15$, $BC = 14$ and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to $\overline{AB}$. If ray $CD$ meets $\overline{AE}$ at $F$, compute $AE \cdot AF$.
[i]Proposed by David Stoner[/i]
1970 All Soviet Union Mathematical Olympiad, 135
The angle bisector $[AD]$, the median $[BM]$ and the height $[CH]$ of the acute-angled triangle $ABC$ intersect in one point. Prove that the $\angle BAC> 45^o$.
2017 China Team Selection Test, 3
Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
Croatia MO (HMO) - geometry, 2016.7
Let $P$ be a point inside a triangle $ABC$ such that
$$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$
Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.
2019 Yasinsky Geometry Olympiad, p2
The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD =
10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.