Found problems: 25757
2008 Vietnam Team Selection Test, 1
On the plane, given an angle $ xOy$. $ M$ be a mobile point on ray $ Ox$ and $ N$ a mobile point on ray $ Oy$. Let $ d$ be the external angle bisector of angle $ xOy$ and $ I$ be the intersection of $ d$ with the perpendicular bisector of $ MN$. Let $ P$, $ Q$ be two points lie on $ d$ such that $ IP \equal{} IQ \equal{} IM \equal{} IN$, and let $ K$ the intersection of $ MQ$ and $ NP$.
$ 1.$ Prove that $ K$ always lie on a fixed line.
$ 2.$ Let $ d_1$ line perpendicular to $ IM$ at $ M$ and $ d_2$ line perpendicular to $ IN$ at $ N$. Assume that there exist the intersections $ E$, $ F$ of $ d_1$, $ d_2$ from $ d$. Prove that $ EN$, $ FM$ and $ OK$ are concurrent.
1978 Vietnam National Olympiad, 5
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance $a$ apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length $c$ and negligible width which can pass through the bend?
2018 Romania Team Selection Tests, 2
Let $ABC$ be a triangle, let $I$ be its incenter, let $\Omega$ be its circumcircle, and let $\omega$ be the $A$- mixtilinear incircle. Let $D,E$ and $T$ be the intersections of $\omega$ and $AB,AC$ and $\Omega$, respectively, let the line $IT$ cross $\omega$ again at $P$, and let lines $PD$ and $PE$ cross the line $BC$ at $M$ and $N$ respectively. Prove that points $D,E,M,N$ are concyclic. What is the center of this circle?
2015 Israel National Olympiad, 2
A triangle is given whose altitudes' lengths are $\frac{1}{5},\frac{1}{5},\frac{1}{8}$. Evaluate the triangle's area.
2006 All-Russian Olympiad, 6
Let $K$ and $L$ be two points on the arcs $AB$ and $BC$ of the circumcircle of a triangle $ABC$, respectively, such that $KL\parallel AC$. Show that the incenters of triangles $ABK$ and $CBL$ are equidistant from the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$.
2023 Sharygin Geometry Olympiad, 10.3
Let $\omega$ be the circumcircle of triangle $ABC$, $O$ be its center, $A'$ be the point of $\omega$ opposite to $A$, and $D$ be a point on a minor arc $BC$ of $\omega$. A point $D'$ is the reflection of $D$ about $BC$. The line $A'D'$ meets for the second time at point $E$. The perpendicular bisector to $D'E$ meets $AB$ and $AC$ at points $F$ and $G$ respectively. Prove that $\angle FOG = 180^\circ - 2\angle BAC$.
1991 Swedish Mathematical Competition, 6
Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.
2000 Brazil Team Selection Test, Problem 3
Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠ $AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.
1990 Dutch Mathematical Olympiad, 4
If $ ABCDEFG$ is a regular $ 7$-gon with side $ 1$, show that: $ \frac{1}{AC}\plus{}\frac{1}{AD}\equal{}1$.
1969 IMO Shortlist, 53
$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$
2022 Novosibirsk Oral Olympiad in Geometry, 5
Two equal rectangles of area $10$ are arranged as follows. Find the area of the gray rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png[/img]
2022 Sharygin Geometry Olympiad, 8.2
Let $ABCD$ be a right-angled trapezoid and $M$ be the midpoint of its greater lateral side $CD$. Circumcircles $\omega_{1}$ and $\omega_{2}$ of triangles $BCM$ and $AMD$ meet for the second time at point $E$. Let $ED$ meet $\omega_{1}$ at point $F$, and $FB$ meet $AD$ at point $G$. Prove that $GM$ bisects angle $BGD$.
2018 Baltic Way, 9
Olga and Sasha play a game on an infinite hexagonal grid. They take turns in placing a stone on a free hexagon of their choice. Olga starts the game. Just before the $2018$th stone is placed, a new rule comes into play. A stone may now be placed only on those free hexagons having at least two occupied neighbors.
A player loses when she or he either is unable to make a move, or makes a move such that a pattern of the rhomboid shape as shown (rotated in any possible way) appears. Determine which player, if any, possesses a winning strategy.
2014 Moldova Team Selection Test, 3
Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.
1984 AMC 12/AHSME, 18
A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y = 2$. Then $x$ is
A. $\sqrt{2} - 1$
B. $\frac{1}{2}$
C. $2 - \sqrt{2}$
D. 1
E. Not uniquely determined
2012 Harvard-MIT Mathematics Tournament, 1
$ABC$ is an isosceles triangle such that $\angle{ABC}=90^\circ$ and $AB=2$. $D$ is the midpoint of $BC$ and $E$ is on $AC$ such that the area of $AEDB$ is twice the area of $ECD$. Find the length of $DE$.
1954 Poland - Second Round, 1
The cross-section of a ball bearing consists of two concentric circles $ C $ and $ C_1 $, between which there are $ n $ small circles $ k_1, k_2, \ldots, k_n $, each of which is tangent to the two adjacent circles and to both circles $ C $ and $ C_1 $. Given the radius $ r $ of the inner circle $ C $ and a natural number $ n $, calculate the radius $ x $ of circle $ C_2 $ passing through the points of tangency of circles $ k_1, k_2, \ldots, k_n $ and the sum $ s $ of the lengths of the arcs of circles $ k_1, k_2, \ldots, k_n $ that lie outside circle $ C_2 $.
2016 Azerbaijan Junior Mathematical Olympiad, 7
In $\triangle ABC$ the median $AM$ is drawn. The foot of perpendicular from $B$ to the angle bisector of $\angle BMA$ is $B_1$ and the foot of perpendicular from $C$ to the angle bisector of $\angle AMC$ is $C_1.$ Let $MA$ and $B_1C_1$ intersect at $A_1.$ Find $\frac{B_1A_1}{A_1C_1}.$
2024 Brazil Team Selection Test, 3
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$.
Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$.
[i]Ivan Chan Kai Chin, Malaysia[/i]
1995 Belarus Team Selection Test, 2
Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$
1999 Gauss, 15
A box contains 36 pink, 18 blue, 9 green, 6 red, and 3 purple cubes that are identical in size. If a cube is selected at random, what is the probability that it is green?
$\textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{8} \qquad \textbf{(C)}\ \dfrac{1}{5} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{9}{70}$
2021 Puerto Rico Team Selection Test, 2
Let $ABC$ be a right triangle with right angle at $ B$ and $\angle C=30^o$. If $M$ is midpoint of the hypotenuse and $I$ the incenter of the triangle, show that $ \angle IMB=15^o$.
2014 Dutch BxMO/EGMO TST, 3
In triangle $ABC$, $I$ is the centre of the incircle. There is a circle tangent
to $AI$ at $I$ which passes through $B$. This circle intersects $AB$ once more
in $P$ and intersects $BC$ once more in $Q$. The line $QI$ intersects $AC$ in $R$.
Prove that $|AR|\cdot |BQ|=|P I|^2$
2011 Sharygin Geometry Olympiad, 8
Given a sheet of tin $6\times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?
2008 China Western Mathematical Olympiad, 1
Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.