Found problems: 25757
2017 International Olympic Revenge, 3
Let $ABC$ be a triangle, and let $P$ be a distinct point on the plane. Moreover, let $A'B'C'$ be a homothety of $ABC$ with ratio $2$ and center $P$, and let $O$ and $O'$ be the circumcenters of $ABC$ and $A'B'C'$, respectively. The circumcircles of $AB'C'$, $A'BC'$, and $A'B'C$ meet at points $X$, $Y$, and $Z$, different from $A'$, $B'$, and $C'$. In a similar way, the circumcircles of $A'BC$, $AB'C$, and $ABC'$ meet at $X'$, $Y'$, and $Z'$, different from $A$, $B$, $C$. Let $W$ and $W'$ be the circumcenters of $XYZ$ and $X'Y'Z'$, respectively. Prove that $OW$ is parallel to $O'W'$.
[i]Proposed by Mateus Thimóteo, Brazil.[/i]
Cono Sur Shortlist - geometry, 1993.14
Prove that the sum of the squares of the distances from a point $P$ to the vertices of a triangle $ABC$ is minimum when $ P$ is the centroid of the triangle.
2015 Dutch IMO TST, 1
In a quadrilateral $ABCD$ we have $\angle A = \angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\angle ADB = \angle EDC$ if and only if $|MA| = |MC|$.
2010 Germany Team Selection Test, 2
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
2019 PUMaC Team Round, 5
Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.
Ukrainian TYM Qualifying - geometry, 2015.20
What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?
2019 CHMMC (Fall), 9
Consider a rectangle with length $6$ and height $4$. A rectangle with length $3$ and height $1$ is placed inside the larger rectangle such that it is distance $1$ from the bottom and leftmost sides of the larger rectangle.
We randomly select one point from each side of the larger rectangle, and connect these $4$ points to form a quadrilateral. What is the probability that the smaller rectangle is strictly contained within that quadrilateral?
2014 IFYM, Sozopol, 4
A square with a side 1 is colored in 3 colors. What’s the greatest real number $a$ such that there can always be found 2 points of the same color at a distance $a$?
2000 Moldova National Olympiad, Problem 8
A rectangular parallelepiped has dimensions $a,b,c$ that satisfy the relation $3a+4b+10c=500$, and the length of the main diagonal $20\sqrt5$. Find the volume and the total area of the surface of the parallelepiped.
2010 China Northern MO, 6
Let $\odot O$ be the inscribed circle of $\vartriangle ABC$, with $D$, $E$, $N$ the touchpoints with sides $AB$, $AC$, $BC$ respectively. Extension of $NO$ intersects segment $DE$ at point $K$. Extension of $AK$ intersects segment $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$.
[img]https://cdn.artofproblemsolving.com/attachments/a/6/a503c500178551ddf9bdb1df0805ed22bc417d.png[/img]
2017 Junior Balkan MO, 3
Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.
2012 Indonesia TST, 3
Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.
2001 Junior Balkan MO, 2
Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$.
[i]Bulgaria[/i]
2019 Sharygin Geometry Olympiad, 2
A point $M$ inside triangle $ABC$ is such that $AM=AB/2$ and $CM=BC/2$. Points $C_0$ and $A_0$ lying on $AB$ and $CB$ respectively are such that $BC_0:AC_0 = BA_0:CA_0 = 3$. Prove that the distances from $M$ to $C_0$ and $A_0$ are equal.
1996 Moldova Team Selection Test, 2
Circles $S_1{}$ and $S_2{}$ intersect in $M{}$ and $N{}$. Line $l$ intersects the circles in points $A,B\in S_1$ and $C,D\in S_2$. Prove that $\angle AMC=\angle BND$ and $\angle ANC=\angle BMD$ if the order of points on line $l$ is:
[b]a)[/b] $A,C,B,D;\quad$ [b]b)[/b] $A,C,D,B.$
2012 Turkmenistan National Math Olympiad, 5
Let $O$ be the center of $\bigtriangleup ABC$'s circumcircle. $CO$ line intersect $AB$ at $D$ and $BO$ line intersect $AC$ at $E$. If $\angle A=\angle CDE=50$° then find $\angle ADE$
1983 IMO Shortlist, 25
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$
2014 BMT Spring, 3
Consider an isosceles triangle $ABC$ ($AB = BC$). Let $D$ be on $BC$ such that $AD \perp BC$ and $O$ be a circle with diameter $BC$. Suppose that segment $AD$ intersects circle $O$ at $E$. If $CA = 2$ what is $CE$?
2005 Harvard-MIT Mathematics Tournament, 5
A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?
2015 IMC, 5
Let $n\ge2$, let $A_1,A_2,\ldots,A_{n+1}$ be $n+1$ points in the
$n$-dimensional Euclidean space, not lying on the same hyperplane,
and let $B$ be a point strictly inside the convex hull of
$A_1,A_2,\ldots,A_{n+1}$. Prove that $\angle A_iBA_j>90^\circ$ holds
for at least $n$ pairs $(i,j)$ with $\displaystyle{1\le i<j\le
n+1}$.
Proposed by Géza Kós, Eötvös University, Budapest
2009 Belarus Team Selection Test, 2
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.
[i]Proposed by Davood Vakili, Iran[/i]
1994 Korea National Olympiad, Problem 3
Let $\alpha,\beta ,\gamma$ be the angles of $\triangle ABC$.
a) Show that $cos^2\alpha +cos^2\beta +cos^2 \gamma =1-2cos\alpha cos\beta cos\gamma$ .
b) Given that $cos\alpha : cos\beta : cos\gamma = 39 : 33 : 25$, find $sin\alpha : sin\beta : sin\gamma$ .
2012 Albania Team Selection Test, 2
It is given an acute triangle $ABC$ , $AB \neq AC$ where the feet of altitude from $A$ its $H$. In the extensions of the sides $AB$ and $AC$ (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.
Find the ratio $\tfrac{HP}{HA}$.
IV Soros Olympiad 1997 - 98 (Russia), 10.1
Two sides of the cyclic quadrilateral $ABCD$ are known: $AB = a$, $BC = b$. A point $K$ is taken on the side $CD$ so that $CK = m$. A circle passing through $B$, $K$ and $D$ intersects line $DA$ at a point $M$, different from $D$. Find $AM$.
2020 Portugal MO, 2
In a triangle $[ABC]$, $\angle C = 2\angle A$. A point $D$ is marked on the side $[AC]$ such that $\angle ABD = \angle DBC$. Knowing that $AB = 10$ and $CD = 3$, what is the length of the side $[BC]$?