Found problems: 25757
2004 Iran MO (3rd Round), 7
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$.
Find the smallest k that:
$S(F) \leq k.P(F)^2$
2023 Stanford Mathematics Tournament, 10
Let $\vartriangle ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, and $CA = 15$. The angle bisector of $\angle BAC$, the angle bisector of $\angle ABC$, and the angle bisector of $\angle ACB$ intersect the circumcircle of $\vartriangle ABC$ again at points $D$, $E$ and $F$, respectively. Compute the area of hexagon $AF BDCE$.
2019 Belarus Team Selection Test, 2.2
Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$.
Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$.
[i](M. Berindeanu, RMC 2018 book)[/i]
1997 Baltic Way, 11
On two parallel lines, the distinct points $A_1,A_2,A_3,\ldots $ respectively $B_1,B_2,B_3,\ldots $ are marked in such a way that $|A_iA_{i+1}|=1$ and $|B_iB_{i+1}|=2$ for $i=1,2,\ldots $. Provided that $A_1A_2B_1=\alpha$, find the infinite sum $\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots $
1999 Turkey Team Selection Test, 2
Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.
1972 Bulgaria National Olympiad, Problem 5
In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.
(a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed.
(b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold?
[i]H. Lesov[/i]
2019 Sharygin Geometry Olympiad, 22
Let $AA_0$ be the altitude of the isosceles triangle $ABC~(AB = AC)$. A circle $\gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$. Let $X$ be an arbitrary point of line $BC$. Prove that the tangents from $X$ to $\gamma$ cut congruent segments on lines $AB$ and $AC$
2000 Federal Competition For Advanced Students, Part 2, 2
A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chose on the arc $ADCB$ in the order $A-P -Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. Show that points $P,Q,X, Y$ lie on a circle.
2017 BMT Spring, 1
What is the largest $n$ such that there exists a non-degenerate convex $n$-gon such that each of its angles are an integer number of degrees, and are all distinct?
2024 Oral Moscow Geometry Olympiad, 3
An equilateral triangle $ABE$ is built inside the square $ABCD$ on the side $AB$, and an equilateral triangle $AFC$ is built on the diagonal $AC$ ($D$ is inside this triangle). The segment $EF$ intersects $CD$ at point $P$. Prove that the lines $AP$, $BE$ and $CF$ intersect at the same point.
2006 MOP Homework, 3
There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps.
what does barycenter of n distinct points mean?
2021 Argentina National Olympiad, 3
Let $ABCD$ be a quadrilateral inscribed in a circle such that $\angle ABC=60^{\circ}.$
a) Prove that if $BC=CD$ then $AB= CD+DA.$
b) Is it true that if $AB= CD+DA$ then $BC=CD$?
2012 China Team Selection Test, 2
Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
1999 IMO Shortlist, 8
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
1988 Mexico National Olympiad, 6
Consider two fixed points $B,C$ on a circle $w$. Find the locus of the incenters of all triangles $ABC$ when point $A$ describes $w$.
2012 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle. Let $BE$ and $CF$ be internal angle bisectors of $\angle B$ and $\angle C$
respectively with $E$ on $AC$ and $F$ on $AB$. Suppose $X$ is a point on the segment $CF$
such that $AX$ perpendicular $CF$; and $Y$ is a point on the segment $BE$ such that $AY$ perpendicular $BE$. Prove
that $XY = (b + c-a)/2$ where $BC = a, CA = b $and $AB = c$.
2014 ELMO Shortlist, 4
Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent.
[i]Proposed by Robin Park[/i]
2004 Junior Balkan Team Selection Tests - Romania, 1
Let $ABC$ be a triangle, having no right angles, and let $D$ be a point on the side $BC$. Let $E$ and $F$ be the feet of the perpendiculars drawn from the point $D$ to the lines $AB$ and $AC$ respectively. Let $P$ be the point of intersection of the lines $BF$ and $CE$. Prove that the line $AP$ is the altitude of the triangle $ABC$ from the vertex $A$ if and only if the line $AD$ is the angle bisector of the angle $CAB$.
2011-2012 SDML (High School), 14
How many numbers among $1,2,\ldots,2012$ have a positive divisor that is a cube other than $1$?
$\text{(A) }346\qquad\text{(B) }336\qquad\text{(C) }347\qquad\text{(D) }251\qquad\text{(E) }393$
Ukraine Correspondence MO - geometry, 2005.4
The bisectors of the angles $A$ and $B$ of the triangle $ABC$ intersect the sides $BC$ and $AC$ at points $D$ and $E$. It is known that $AE + BD = AB$. Find the angle $\angle C$.
2014 Contests, 2
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.
[i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]
1980 Tournament Of Towns, (005) 5
A finite set of line segments, of total length $18$, belongs to a square of unit side length (we assume that the square includes its boundary and that a line segment includes its end points). The line segments are parallel to the sides of the square and may intersect one another. Prove that among the regions into which the square is divided by the line segments, at least one of these must have area not less than $0.01$.
(A Berzinsh, Riga)
2012 NZMOC Camp Selection Problems, 1
From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?
1995 China National Olympiad, 1
Given four spheres with their radii equal to $2,2,3,3$ respectively, each sphere externally touches the other spheres. Suppose that there is another sphere that is externally tangent to all those four spheres, determine the radius of this sphere.
2014 Germany Team Selection Test, 3
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.