Found problems: 25757
1957 Poland - Second Round, 2
Prove that if $ M $, $ N $, $ P $ are the feet of the altitudes of acute-angled triangle $ ABC $, then the ratio of the perimeter of triangle $ MNP $ to the perimeter of triangle $ ABC $ is equal to the ratio of the radius of the circle inscribed in triangle $ ABC $ to the radius of the circle circumscribed about triangle $ ABC $.
2013 Baltic Way, 14
Circles $\alpha$ and $\beta$ of the same radius intersect in two points, one of which is $P$. Denote by $A$ and $B$, respectively, the points diametrically opposite to $P$ on each of $\alpha$ and $\beta$ . A third circle of the same radius passes through $P$ and intersects $\alpha$ and $\beta$ in the points $X$ and $Y$ , respectively. Show that the line $XY$ is parallel to the line $AB$.
2022 Poland - Second Round, 4
Given quadrilateral $ABCD$ inscribed into a circle with diagonal $AC$ as diameter. Let $E$ be a point on segment $BC$ s.t. $\sphericalangle DAC=\sphericalangle EAB$. Point $M$ is midpoint of $CE$. Prove that $BM=DM$.
1971 Czech and Slovak Olympiad III A, 6
Let a tetrahedron $ABCD$ and its inner point $O$ be given. For any edge $e$ of $ABCD$ consider the segment $f(e)$ containing $O$ such that $f(e)\parallel e$ and the endpoints of $f(e)$ lie on the faces of the tetrahedron. Show that \[\sum_{e\text{ edge}}\,\frac{\,f(e)\,}{e}=3.\]
1988 IMO Longlists, 68
In a group of $n$ people, each one knows exactly three others. They are seated around a table. We say that the seating is $perfect$ if everyone knows the two sitting by their sides. Show that, if there is a perfect seating $S$ for the group, then there is always another perfect seating which cannot be obtained from $S$ by rotation or reflection.
2004 Germany Team Selection Test, 3
Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$.
(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.
(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$.
[i]Radu Gologan, Romania[/i]
[hide="Remark"]
The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url].
[/hide]
2002 Polish MO Finals, 2
On sides $AC$ and $BC$ of acute-angled triangle $ABC$ rectangles with equal areas $ACPQ$ and $BKLC$ were built exterior. Prove that midpoint of $PL$, point $C$ and center of circumcircle are collinear.
1994 China National Olympiad, 1
Let $ABCD$ be a trapezoid with $AB\parallel CD$. Points $E,F$ lie on segments $AB,CD$ respectively. Segments $CE,BF$ meet at $H$, and segments $ED,AF$ meet at $G$. Show that $S_{EHFG}\le \dfrac{1}{4}S_{ABCD}$. Determine, with proof, if the conclusion still holds when $ABCD$ is just any convex quadrilateral.
2008 Oral Moscow Geometry Olympiad, 6
Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$.
(A. Zaslavsky)
2024 BMT, 2
On a chalkboard, Benji draws a square with side length $6.$ He then splits each side into $3$ equal segments using $2$ points for a total of $12$ segments and $8$ points. After trying some shapes, Benji finds that by using a circle, he can connect all $8$ points together. What is the area of this circle?
[center]
[img]
https://cdn.artofproblemsolving.com/attachments/d/f/ead2d0c85f474612d49d3a023a43215e11066b.png
[/img]
[/center]
1993 Vietnam National Olympiad, 2
$ABCD$ is a quadrilateral such that $AB$ is not parallel to $CD$, and $BC$ is not parallel to $AD$. Variable points $P, Q, R, S$ are taken on $AB, BC, CD, DA$ respectively so that $PQRS$ is a parallelogram. Find the locus of its center.
2011 AMC 10, 11
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
$\textbf{(A)}\,\frac{49}{64} \qquad\textbf{(B)}\,\frac{25}{32} \qquad\textbf{(C)}\,\frac78 \qquad\textbf{(D)}\,\frac{5\sqrt{2}}{8} \qquad\textbf{(E)}\,\frac{\sqrt{14}}{4} $
1940 Moscow Mathematical Olympiad, 060
Construct a circle equidistant from four points on a plane. How many solutions are there?
2021 New Zealand MO, 5
Let $ABC$ be an isosceles triangle with $AB = AC$. Point $D$ lies on side $AC$ such that $BD$ is the angle bisector of $\angle ABC$. Point $E$ lies on side $BC$ between $B$ and $C$ such that $BE = CD$. Prove that $DE$ is parallel to $AB$.
2008 Junior Balkan Team Selection Tests - Moldova, 7
In an acute triangle $ABC$, points $A_1, B_1, C_1$ are the midpoints of the sides $BC, AC, AB$, respectively. It is known that $AA_1 = d(A_1, AB) + d(A_1, AC)$, $BB1 = d(B_1, AB) + d(A_1, BC)$, $CC_1 = d(C_1, AC) + d(C_1, BC)$, where $d(X, Y Z)$ denotes the distance from point $X$ to the line $YZ$. Prove, that triangle $ABC$ is equilateral.
2017 CCA Math Bonanza, T7
Let $ABCD$ be a convex quadrilateral with $AC=20$, $BC=12$ and $BD=17$. If $\angle{CAB}=80^{\circ}$ and $\angle{DBA}=70^{\circ}$, then find the area of $ABCD$.
[i]2017 CCA Math Bonanza Team Round #7[/i]
1998 Dutch Mathematical Olympiad, 4
Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$.
(a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$.
(b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.
2006 Austrian-Polish Competition, 3
$ABCD$ is a tetrahedron.
Let $K$ be the center of the incircle of $CBD$.
Let $M$ be the center of the incircle of $ABD$.
Let $L$ be the gravycenter of $DAC$.
Let $N$ be the gravycenter of $BAC$.
Suppose $AK$, $BL$, $CM$, $DN$ have one common point.
Is $ABCD$ necessarily regular?
2019 Oral Moscow Geometry Olympiad, 1
Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.
2000 All-Russian Olympiad Regional Round, 9.7
On side $AB$ of triangle $ABC$, point $D$ is selected. Circle circumscribed around triangle $BCD$, intersects side $AC$ at point $M$, and the circumcircle of triangle $ACD$ intersects the side $BC$ at point $ N$ ($M,N \ne C$). Let $O$ be the circumcenter of the triangle $CMN$. Prove that line $OD$ is perpendicular to side $AB$.
1993 India Regional Mathematical Olympiad, 3
Suppose $A_1, A_2, A_3, \ldots, A_{20}$is a 20 sides regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but the sides are not the sides of the polygon?
1986 IMO Longlists, 41
Let $M,N,P$ be the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$. The lines $AM, BN, CP$ intersect the circumcircle of $ABC$ at points $A',B', C'$, respectively. Show that if $A'B'C'$ is an equilateral triangle, then so is $ABC.$
2013 ELMO Shortlist, 1
Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral.
[i]Proposed by Owen Goff[/i]
2020 IMO, 1
Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold:
\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]
Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$.
[i]Proposed by Dominik Burek, Poland[/i]
1999 French Mathematical Olympiad, Problem 1
What is the maximum possible volume of a cylinder inscribed in a cone and having the same axis of symmetry as the cone? What is the maximum possible volume of a ball inscribed in the cone with center on the axis of symmetry of the cone? Compare these three volumes.