Found problems: 25757
2013 ELMO Shortlist, 6
Let $ABCDEF$ be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define $X=AB\cap DE$, $Y=BC\cap EF$, and $Z=CD\cap FA$. Prove that
\[\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.\][i]Proposed by Victor Wang[/i]
2008 Hungary-Israel Binational, 1
Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.
2005 Paraguay Mathematical Olympiad, 2
If you multiply the number of faces that a pyramid has with the number of edges of the pyramid, you get $5.100$. Determine the number of faces of the pyramid.
1997 Czech and Slovak Match, 4
Is it possible to place $100$ balls in space so that no two of them have a common interior point and each of them touches at least one third of the others?
2003 Kazakhstan National Olympiad, 3
Two square sheets have areas equal to $ 2003$. Each of the sheets is arbitrarily divided into $ 2003$ nonoverlapping polygons, besides, each of the polygons has an unitary area. Afterward, one overlays two sheets, and it is asked to prove that the obtained double layer can be punctured $ 2003$ times, so that each of the $ 4006$ polygons gets punctured precisely once.
2007 All-Russian Olympiad, 2
The numbers $1,2,\ldots,100$ are written in the cells of a $10\times 10$ table, each number is written once. In one move, Nazar may interchange numbers in any two cells. Prove that he may get a table where the sum of the numbers in every two adjacent (by side) cells is composite after at most $35$ such moves.
[i]N. Agakhanov[/i]
2004 Cono Sur Olympiad, 2
Given a circle $C$ and a point $P$ on its exterior, two tangents to the circle are drawn through $P$, with $A$ and $B$ being the points of tangency. We take a point $Q$ on the minor arc $AB$ of $C$. Let $M$ be the intersection of $AQ$ with the line perpendicular to $AQ$ that goes through $P$, and let $N$ be the intersection of $BQ$ with the line perpendicular to $BQ$ that goes through $P$.
Show that, by varying $Q$ on the minor arc $AB$, all of the lines $MN$ pass through the same point.
2022 Turkey Junior National Olympiad, 4
In parallellogram $ABCD$, on the arc $BC$ of the circumcircle $(ABC)$, not containing the point $A$, we take a point $P$ and on the $[AC$, we take a point $Q$ such that $\angle PBC= \angle CDQ$. Prove that $(APQ)$ is tangent to $AB$.
1990 IMO Longlists, 69
Consider the set of cuboids: the three edges $a, b, c$ from a common vertex satisfy the condition
\[\frac ab = \frac{a^2}{c^5}\]
(i) Prove that there are $100$ pairs of cuboids in this set with equal volumes in each pair.
(ii) For each pair of the above cuboids, find the ratio of the sum of their edges.
1952 Moscow Mathematical Olympiad, 212
Prove that if the orthocenter divides all heights of a triangle in the same proportion, the triangle is equilateral.
2018 Istmo Centroamericano MO, 5
Let $ABC$ be an isosceles triangle with $CA = CB$. Let $D$ be the foot of the alttiude from $C$, and $\ell$ be the external angle bisector at $C$. Take a point $N$ on $\ell$ so that $AN> AC$ , on the same side as $A$ wrt $CD$. The bisector of the angle $NAC$ cuts $\ell$'at $F$. Show that $\angle NCD + \angle BAF> 180^o.$
2015 Iran Geometry Olympiad, 3
In the figure below, we know that $AB = CD$ and $BC = 2AD$. Prove that $\angle BAD = 30^o$.
[img]https://3.bp.blogspot.com/-IXi_8jSwzlU/W1R5IydV5uI/AAAAAAAAIzo/2sREnDEnLH8R9zmAZLCkVCGeMaeITX9YwCK4BGAYYCw/s400/IGO%2B2015.el3.png[/img]
2012 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle. Let $E$ be a point on the segment $BC$ such that $BE = 2EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AE$ in $Q$. Determine $BQ:QF$.
2000 Harvard-MIT Mathematics Tournament, 5
Find the interior angle between two sides of a regular octagon (degrees).
1992 Poland - First Round, 10
Let $C$ be a cube and let $f: C \longrightarrow C$ be a surjection with
$|PQ| \geq |f(P)f(Q)|$
for all $P,Q \in C$. Prove that $f$ is an isometry.
Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.41
Let $AC$ be the largest side of the triangle $ABC$. The point M is selected on the ray $AC$ ray, and point $N$ on ray $CA$ such that $CN = CB$ and$ AM = AB$ .
a) Prove that $\vartriangle ABC$ is isosceles if we know that $BM = BN$.
b) Will the statement remain true if $AC$ is not necessarily the largest side of triangle $ABC$?
2004 All-Russian Olympiad, 2
Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.
2007 Switzerland - Final Round, 4
Let $ABC$ be an acute-angled triangle with $AB> AC$ and orthocenter $H$. Let $D$ the projection of $A$ on $BC$. Let $E$ be the reflection of $C$ wrt $D$. The lines $AE$ and $BH$ intersect at point $S$. Let $N$ be the midpoint of $AE$ and let $M$ be the midpoint of $BH$. Prove that $MN$ is perpendicular to $DS$.
2015 Taiwan TST Round 3, 2
Let $O$ be the circumcircle of the triangle $ABC$. Two circles $O_1,O_2$ are tangent to each of the circle $O$ and the rays $\overrightarrow{AB},\overrightarrow{AC}$, with $O_1$ interior to $O$, $O_2$ exterior to $O$. The common tangent of $O,O_1$ and the common tangent of $O,O_2$ intersect at the point $X$. Let $M$ be the midpoint of the arc $BC$ (not containing the point $A$) on the circle $O$, and the segment $\overline{AA'}$ be the diameter of $O$. Prove that $X,M$, and $A'$ are collinear.
2002 Croatia National Olympiad, Problem 2
Consider the cube with the vertices $A(1,1,1)$, $B(-1,1,1)$, $C(-1,-1,1)$, $D(1,-1,1)$ and $A',B',C',D'$ symmetric to $A,B,C,D$ respectively with respect to the origin $O$. Let $T$ be a point not on the circumsphere of the cube and let $OT=d$. Denote $\alpha=\angle ATA'$, $\beta=\angle BTB'$, $\gamma=\angle CTC'$, $\delta=\angle DTD'$. Prove that
$$\tan^2\alpha+\tan^2\beta+\tan^2\gamma+\tan^2\delta=\frac{32d^2}{\left(d^2-3\right)^2}.$$
2018 Harvard-MIT Mathematics Tournament, 2
Points $A,B,C,D$ are chosen in the plane such that segments $AB,BC,CD,DA$ have lengths $2,7,5,12,$ respectively. Let $m$ be the minimum possible value of the length of segment $AC$ and let $M$ be the maximum possible value of the length of segment $AC.$ What is the ordered pair $(m,M)$?
2014 Dutch IMO TST, 2
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.
2016 Sharygin Geometry Olympiad, 7
Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point.
[i](The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)[/i]
1967 IMO, 4
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$
2024 Sharygin Geometry Olympiad, 3
Let $ABC$ be an acute-angled triangle, and $M$ be the midpoint of the minor arc $BC$ of its circumcircle. A circle $\omega$ touches the side $AB, AC$ at points $P, Q$ respectively and passes through $M$. Prove that $BP + CQ = PQ$.