Found problems: 25757
Novosibirsk Oral Geo Oly VIII, 2017.6
In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.
2010 Polish MO Finals, 3
$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.
2019 Lusophon Mathematical Olympiad, 3
Let $ABC$ be a triangle with $AC \ne BC$. In triangle $ABC$, let $G$ be the centroid, $I$ the incenter and O Its circumcenter. Prove that $IG$ is parallel to $AB$ if, and only if, $CI$ is perpendicular on $IO$.
2000 Nordic, 3
In the triangle $ABC$, the bisector of angle $\angle B$ meets $AC$ at $D$ and the bisector of angle $\angle C$ meets $AB$ at $E$. The bisectors meet each other at $O$. Furthermore, $OD = OE$. Prove that either $ABC$ is isosceles or $\angle BAC = 60^\circ$.
2020 Jozsef Wildt International Math Competition, W58
In all triangles $ABC$ does it hold that:
$$\sum\sqrt{\frac{a(h_a-2r)}{(3a+b+c)(h_a+2r)}}\le\frac34$$
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
Ukrainian TYM Qualifying - geometry, I.13
A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?
2002 Italy TST, 1
Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.
2007 Tuymaada Olympiad, 3
$ AA_{1}$, $ BB_{1}$, $ CC_{1}$ are altitudes of an acute triangle $ ABC$. A circle passing through $ A_{1}$ and $ B_{1}$ touches the arc $ AB$ of its circumcircle at $ C_{2}$. The points $ A_{2}$, $ B_{2}$ are defined similarly. Prove that the lines $ AA_{2}$, $ BB_{2}$, $ CC_{2}$ are concurrent.
2002 India National Olympiad, 1
For a convex hexagon $ ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements.
($ a_1$) $ AB$ is parallel to $ DE$. ($ a_2$)$ AE \equal{} BD$.
($ b_1$) $ BC$ is parallel to $ EF$. ($ b_2$)$ BF \equal{} CE$.
($ c_1$) $ CD$ is parallel to $ FA$. ($ c_2$) $ CA \equal{} DF$.
$ (a)$ Show that if all six of these statements are true then the hexagon is cyclic.
$ (b)$ Prove that, in fact, five of the six statements suffice.
1996 Korea National Olympiad, 4
Circle $C$(the center is $C$.) is inside the $\angle XOY$ and it is tangent to the two sides of the angle. Let $C_1$ be the circle that passes through the center of $C$ and tangent to two sides of angle and let $A$ be one of the endpoint of diameter of $C_1$ that passes through $C$ and $B$ be the intersection of this diameter and circle $C.$ Prove that the cirlce that $A$ is the center and $AB$ is the radius is also tangent to the two sides of $\angle XOY.$
1998 Portugal MO, 2
The regular octagon of the following figure is inscribed in a circle of radius $1$ and $P$ is a arbitrary point of this circle. Calculate the value of $PA^2 + PB^2 +...+ PH^2$.
[img]https://cdn.artofproblemsolving.com/attachments/4/c/85e8e48c45970556077ac09c843193959b0e5a.png[/img]
2018 Sharygin Geometry Olympiad, 3
A cyclic $n$-gon is given. The midpoints of all its sides are concyclic. The sides of the $n$-gon cut $n$ arcs of this circle lying outside the $n$-gon. Prove that these arcs can be coloured red and blue in such a way that the sum of the lengths of the red arcs is equal to the sum of the lengths of the blue arcs.
2013 Cuba MO, 2
Two equal isosceles triangles $ABC$ and $ADB$, with $C$ and $D$ located in different halfplanes with respect to the line $AB$, share the base $AB$. The midpoints of $AC$ and $BC$ are denoted by $E$ and $F$ respectively. Show that $DE$ and $DF$ divide $AB$ into three equal parts length.
2016 Mathematical Talent Reward Programme, SAQ: P 2
5 blocks of volume 1 cm$^3$, 1 cm$^3$, 1 cm$^3$, 1 cm$^3$ and 4 cm$^3$ are placed one above another to form a structure as shown in the figure. Suppose sum of surface areas of upper face of each is 48 cm$^2$ . Determine the minimum possible height of the whole structure.
2018 Ramnicean Hope, 2
Let be the points $ M,N,P, $ on the sides $ BC,AC,AB $ (not on their endpoints), respectively, of a triangle $ ABC, $ such that $ \frac{BM}{MC} =\frac{CN}{NA} =\frac{AP}{PB} . $ Denote $ G_1,G_2,G_3 $ the centroids of $ APN,BMP,CNM, $ respectively. Show that the $ MNP $ has the same centroid as $ G_1G_2G_3. $
[i]Ovidiu Țâțan[/i]
2009 Balkan MO Shortlist, G4
Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$.
[i]Liubomir Chiriac, Moldova[/i]
2016 Brazil National Olympiad, 1
Let $ABC$ be a triangle.
$r$ and $s$ are the angle bisectors of $\angle ABC$ and $\angle BCA$, respectively.
The points $E$ in $r$ and $D$ in $s$ are such that $AD \| BE$ and $AE \| CD$.
The lines $BD$ and $CE$ cut each other at $F$.
$I$ is the incenter of $ABC$.
Show that if $A,F,I$ are collinear, then $AB=AC$.
2002 Greece National Olympiad, 3
In a triangle $ABC$ we have $\angle C>10^0$ and $\angle B=\angle C+10^0.$We consider point $E$ on side $AB$ such that $\angle ACE=10^0,$ and point $D$ on side $AC$ such that $\angle DBA=15^0.$ Let $Z\neq A$ be a point of interection of the circumcircles of the triangles $ABD$ and $AEC.$Prove that $\angle ZBA>\angle ZCA.$
1965 IMO Shortlist, 6
In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.
1987 IMO Longlists, 52
Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear.
[i]Proposed by Poland.[/i]
BIMO 2022, 1
Given an acute triangle $ABC$, mark $3$ points $X, Y, Z$ in the interior of the triangle. Let $X_1, X_2, X_3$ be the projections of $X$ to $BC, CA, AB$ respectively, and define the points $Y_i, Z_i$ similarly for $i=1, 2, 3$.
a) Suppose that $X_iY_i<X_iZ_i$ for all $i=1,2,3$, prove that $XY<XZ$.
b) Prove that this is not neccesarily true, if triangle $ABC$ is allowed to be obtuse.
[i]Proposed by Ivan Chan Kai Chin[/i]
2024 All-Russian Olympiad, 4
Let $ABCD$ be a convex quadrilateral with $\angle A+\angle D=90^\circ$ and $E$ the point of intersection of its diagonals. The line $\ell$ cuts the segments $AB$, $CD$, $AE$ and $ED$ in points $X,Y,Z,T$, respectively. Suppose that $AZ=CE$ and $BE=DT$. Prove that the length of the segment $XY$ is not larger than the diameter of the the circumcircle of $ETZ$.
[i]Proposed by A. Kuznetsov, I. Frolov[/i]
Oliforum Contest III 2012, 3
Show that if equiangular hexagon has sides $a, b, c, d, e, f$ in order then $a - d = e - b = c - f$.
2021/2022 Tournament of Towns, P3
The intersection of two triangles is a hexagon. If this hexagon is removed, six small triangles remain. These six triangles have the same in-radii. Prove the in-radii of the original two triangles are also equal.
Spoiler:
This is one of the highlights of TT. Also SA3
2014 AMC 8, 14
Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?
[asy]
size(250);
defaultpen(linewidth(0.8));
pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0);
draw(A--B--E--D--cycle^^C--D);
draw(rightanglemark(D,C,E,30));
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,S);
label("$D$",D,N);
label("$E$",E,S);
label("$5$",A/2,W);
label("$6$",(A+D)/2,N);
[/asy]
$\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad \textbf{(E) }16$