Found problems: 25757
1955 Moscow Mathematical Olympiad, 304
The centers $O_1, O_2$ and $O_3$ of circles exscribed about $\vartriangle ABC$ are connected. Prove that $O_1O_2O_3$ is an acute-angled one.
1994 Abels Math Contest (Norwegian MO), 1a
In a half-ball of radius $3$ is inscribed a cylinder with base lying on the base plane of the half-ball, and another such cylinder with equal volume. If the base-radius of the first cylinder is $\sqrt3$, what is the base-radius of the other one?
Kvant 2020, M2596
The circle $\omega{}$ is inscribed in the quadrilateral $ABCD$. Prove that the diameter of the circle $\omega{}$ does not exceed the length of the segment connecting the midpoints of the sides $BC$ and $AD$.
[i]Proposed by O. Yuzhakov[/i]
2017 Moscow Mathematical Olympiad, 5
$8$ points lie on the faces of unit cube and form another cube. What can be length of edge of this cube?
1999 Hong kong National Olympiad, 2
Let $I$ be the incentre and $O$ the circumcentre of a non-equilateral triangle $ABC$. Prove that $\angle AIO \le 90^{\circ}$ if and only if $2BC\le AB+AC$.
2011 IMO Shortlist, 2
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]
[i]Proposed by Alexey Gladkich, Israel[/i]
2012 Kosovo National Mathematical Olympiad, 2
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that,
$\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$
2020 Yasinsky Geometry Olympiad, 1
In the rectangle $ABCD$, $AB = 2BC$. An equilateral triangle $ABE$ is constructed on the side $AB$ of the rectangle so that its sides $AE$ and $BE$ intersect the segment $CD$. Point $M$ is the midpoint of $BE$. Find the $\angle MCD$.
2003 China Team Selection Test, 2
Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$.
Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$.
Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$).
Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$).
Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$.
Find the length of $AP$.
2009 China Western Mathematical Olympiad, 1
Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.
2022 Kazakhstan National Olympiad, 1
Given a triangle $ABC$ draw the altitudes $AD$, $BE$, $CF$. Take points $P$ and $Q$ on $AB$ and $AC$, respectively such that $PQ \parallel BC$. Draw the circles with diameters $BQ$ and $CP$ and let them intersect at points $R$
and $T$ where $R$ is closer to $A$ than $T$. Draw the altitudes $BN$ and $CM$ in the triangle $BCR$. Prove that $FM$, $EN$ and $AD$ are concurrent.\\
1950 Polish MO Finals, 3
Prove that if the two altitudes of a tetrahedron intersect, then the other two atltitudes intersect also.
2005 Indonesia MO, 6
Find all triples $ (x,y,z)$ of integers which satisfy
$ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$
$ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$
$ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.
2016 China Northern MO, 2
In isosceles triangle $ABC$, $\angle CAB=\angle CBA=\alpha$, points $P,Q$ are on different sides of line $AB$, and $\angle CAP=\angle ABQ=\beta,\angle CBP=\angle BAQ=\gamma$. Prove that $P,C,Q$ are colinear.
2016 IFYM, Sozopol, 5
A convex quadrilateral is cut into smaller convex quadrilaterals so that they are adjacent to each other only by whole sides.
a) Prove that if all small quadrilaterals are inscribed in a circle, then the original one is also inscribed in a circle.
b) Prove that if all small quadrilaterals are cyclic, then the original one is also cyclic.
1997 Bundeswettbewerb Mathematik, 3
A semicircle with diameter $AB = 2r$ is divided into two sectors by an arbitrary radius. To each of the sectors a circle is inscribed. These two circles touch A$B$ at $S$ and $T$. Show that $ST \ge 2r(\sqrt{2}-1)$.
Ukrainian TYM Qualifying - geometry, 2018.18
In the acute triangle $ABC$, the altitude $AH$ is drawn. Using segments $AB,BH,CH$ and $AC$ as diameters circles $\omega_1, \omega_2, \omega_3$ and $\omega_4$ are constructed respectively. Besides the point $H$, the circles $\omega_1$ and $\omega_3$ intersect at the point $P,$ and the circles $\omega_2$ and $\omega_4$ interext at point $Q$. The lines $BQ$ and $CP$ intersect at point $N$. Prove that this point lies on the midline of triangle $ABC$, which is parallel to $BC$.
2005 JBMO Shortlist, 6
Let $C_1,C_2$ be two circles intersecting at points $A,P$ with centers $O,K$ respectively. Let $B,C$ be the symmetric of $A$ wrt $O,K$ in circles $C_1,C_2 $ respectively. A random line passing through $A$ intersects circles $C_1,C_2$ at $D,E$ respectively. Prove that the center of circumcircle of triangle $DEP$ lies on the circumcircle of triangle $OKP$.
2011 AMC 10, 7
The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is $30 ^\circ$ larger than the other. What is the degree measure of the largest angle in the triangle?
$ \textbf{(A)}\ 69 \qquad
\textbf{(B)}\ 72 \qquad
\textbf{(C)}\ 90 \qquad
\textbf{(D)}\ 102 \qquad
\textbf{(E)}\ 108 $
Russian TST 2017, P3
Let $K=(V, E)$ be a finite, simple, complete graph. Let $\phi: E \to \mathbb{R}^2$ be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle are collinear. Show that the range of $\phi$ is contained in a line.
2008 Tournament Of Towns, 1
A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.
2011 India National Olympiad, 4
Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
2016 AMC 10, 11
What is the area of the shaded region of the given $8 \times 5$ rectangle?
[asy]
size(6cm);
defaultpen(fontsize(9pt));
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));
label("$1$",(1/2,5),dir(90));
label("$7$",(9/2,5),dir(90));
label("$1$",(8,1/2),dir(0));
label("$4$",(8,3),dir(0));
label("$1$",(15/2,0),dir(270));
label("$7$",(7/2,0),dir(270));
label("$1$",(0,9/2),dir(180));
label("$4$",(0,2),dir(180));
[/asy]
$\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8$
May Olympiad L2 - geometry, 2016.4
In a triangle $ABC$, let $D$ and $E$ be points of the sides $ BC$ and $AC$ respectively. Segments $AD$ and $BE$ intersect at $O$. Suppose that the line connecting midpoints of the triangle and parallel to $AB$, bisects the segment $DE$. Prove that the triangle $ABO$ and the quadrilateral $ODCE$ have equal areas.
2005 Junior Balkan Team Selection Tests - Romania, 5
On the sides $AD$ and $BC$ of a rhombus $ABCD$ we consider the points $M$ and $N$ respectively. The line $MC$ intersects the segment $BD$ in the point $T$, and the line $MN$ intersects the segment $BD$ in the point $U$. We denote by $Q$ the intersection between the line $CU$ and the side $AB$ and with $P$ the intersection point between the line $QT$ and the side $CD$.
Prove that the triangles $QCP$ and $MCN$ have the same area.