Found problems: 25757
2007 Serbia National Math Olympiad, 2
In a scalene triangle $ABC , AD, BE , CF$ are the angle bisectors $(D \in BC , E \in AC , F \in AB)$. Points $K_{a}, K_{b}, K_{c}$ on the incircle of triangle $ABC$ are such that $DK_{a}, EK_{b}, FK_{c}$ are tangent to the incircle and $K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB$. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC , CA, AB$ , respectively. Prove that the lines $A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.
2012 Online Math Open Problems, 5
Congruent circles $\Gamma_1$ and $\Gamma_2$ have radius $2012,$ and the center of $\Gamma_1$ lies on $\Gamma_2.$ Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. The line through $A$ perpendicular to $AB$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$, respectively. Find the length of $CD$.
[i]Author: Ray Li[/i]
1993 Baltic Way, 16
Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$, so that $|AB|=|BC|=|CD|=14\text{cm}$. Another line intersects the circles at $E,F$, respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$. Find the radius $r$.
2000 Brazil Team Selection Test, Problem 3
Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠$AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.
2018 Sharygin Geometry Olympiad, 14
Let $ABC$ be a right-angled triangle with $\angle C = 90^{\circ}$, $K$, $L$, $M$ be the midpoints of sides $AB$, $BC$, $CA$ respectively, and $N$ be a point of side $AB$. The line $CN$ meets $KM$ and $KL$ at points $P$ and $Q$ respectively. Points $S$, $T$ lying on $AC$ and $BC$ respectively are such that $APQS$ and $BPQT$ are cyclic quadrilaterals. Prove that
a) if $CN$ is a bisector, then $CN$, $ML$ and $ST$ concur;
b) if $CN$ is an altitude, then $ST$ bisects $ML$.
1962 All Russian Mathematical Olympiad, 014
Given the circumference $s$ and the straight line $l$, passing through the centre $O$ of $s$. Another circumference $s'$ passes through the point $O$ and has its centre on the $l$. Describe the set of the points $M$, where the common tangent of $s$ and $s'$ touches $s'$.
1991 Tournament Of Towns, (318) 5
Let $M$ be a centre of gravity (the intersection point of the medians) of a triangle $ABC$. Under rotation by $120$ degrees about the point $M$, the point $B$ is taken to the point $P$; under rotation by $240$ degrees about $M$, the point $C$ is taken to the point $Q$. Prove that either $APQ$ is an equilateral triangle, or the points $A, P, Q$ coincide.
(Bykovsky, Khabarovsksk)
2020 CMIMC Geometry, 9
In triangle $ABC$, points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$. Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$. Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that $PB = 5$, compute the length of $BC$.
2016 Costa Rica - Final Round, G1
Let $\vartriangle ABC$ be isosceles with $AB = AC$. Let $\omega$ be its circumscribed circle and $O$ its circumcenter. Let $D$ be the second intersection of $CO$ with $\omega$. Take a point $E$ in $AB$ such that $DE \parallel AC$ and suppose that $AE: BE = 2: 1$. Show that $\vartriangle ABC$ is equilateral.
2014 Balkan MO Shortlist, G7
Let $I$ be the incenter of $\triangle ABC$ and let $H_a$, $H_b$, and $H_c$ be the orthocenters of $\triangle BIC$ , $\triangle CIA$, and $\triangle AIB$, respectively. The lines $H_aH_b$ meets $AB$ at $X$ and the line $H_aH_c$ meets $AC$ at $Y$. If the midpoint $T$ of the median $AM$ of $\triangle ABC$ lies on $XY$, prove that the line $H_aT$ is perpendicular to $BC$
2020 Novosibirsk Oral Olympiad in Geometry, 1
Two semicircles touch the side of the rectangle, each other and the segment drawn in it as in the figure. What part of the whole rectangle is filled?
[img]https://cdn.artofproblemsolving.com/attachments/3/e/70ca8b80240a282553294a58cb3ed807d016be.png[/img]
1983 Vietnam National Olympiad, 3
A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.
1989 China National Olympiad, 1
We are given two point sets $A$ and $B$ which are both composed of finite disjoint arcs on the unit circle. Moreover, the length of each arc in $B$ is equal to $\dfrac{\pi}{m}$ ($m \in \mathbb{N}$). We denote by $A^j$ the set obtained by a counterclockwise rotation of $A$ about the center of the unit circle for $\dfrac{j\pi}{m}$ ($j=1,2,3,\dots$). Show that there exists a natural number $k$ such that $l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)$.(Here $l(X)$ denotes the sum of lengths of all disjoint arcs in the point set $X$)
2008 Peru MO (ONEM), 4
All points in the plane that have both integer coordinates are painted, using the colors red, green, and yellow. If the points are painted so that there is at least one point of each color.
Prove that there are always three points $X$, $Y$ and $Z$ of different colors, such that $\angle XYZ = 45^{\circ} $
1942 Eotvos Mathematical Competition, 1
Prove that in any triangle, at most one side can be shorter than the altitude from the opposite vertex.
2017 India National Olympiad, 6
Let $n\ge 1$ be an integer and consider the sum $$x=\sum_{k\ge 0} \dbinom{n}{2k} 2^{n-2k}3^k=\dbinom{n}{0}2^n+\dbinom{n}{2}2^{n-2}\cdot{}3+\dbinom{n}{4}2^{n-k}\cdot{}3^2 + \cdots{}.$$
Show that $2x-1,2x,2x+1$ form the sides of a triangle whose area and inradius are also integers.
2015 Junior Regional Olympiad - FBH, 3
Let $AD$ be a median of $ABC$ and $S$ its midpoint. Let $E$ be a intersection point of $AB$ and $CS$. Prove that $BE=2AE$
1941 Moscow Mathematical Olympiad, 087
On a plane, several points are chosen so that a disc of radius $1$ can cover every $3$ of them. Prove that a disc of radius $1$ can cover all the points.
1971 IMO Shortlist, 4
We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.
2023 Saint Petersburg Mathematical Olympiad, 2
Given is a triangle $ABC$ with median $BM$. The point $D$ lies on the line $AC$ after $C$, such that $BD=2CD$. The circle $(BMC)$ meets the segment $BD$ at $N$. Show that $AC+BM>2MN$.
2023 Purple Comet Problems, 15
A rectangle with integer side lengths has the property that its area minus $5$ times its perimeter equals $2023$. Find the minimum possible perimeter of this rectangle.
1998 Croatia National Olympiad, Problem 1
Let there be a given parabola $y^2=4ax$ in the coordinate plane. Consider all chords of the parabola that are visible at a right angle from the origin of the coordinate system. Prove that all these chords pass through a fixed point.
Denmark (Mohr) - geometry, 2000.2
Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom?
[img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]
2012 Turkey MO (2nd round), 6
Let $B$ and $D$ be points on segments $[AE]$ and $[AF]$ respectively. Excircles of triangles $ABF$ and $ADE$ touching sides $BF$ and $DE$ is the same, and its center is $I$. $BF$ and $DE$ intersects at $C$. Let $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$ be the circumcenters of triangles $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$ respectively.
[b]a) [/b] Show that points $P_1, P_2, P_3, P_4$ concylic and points $Q_1, Q_2, Q_3, Q_4$ concylic.
[b]b) [/b] Denote centers of theese circles as $O_1$ and $O_2$. Prove that $O_1, O_2$ and $I$ are collinear.
2024 Korea Junior Math Olympiad (First Round), 16.
There is an Equilateral trapezoid $ ABCD. $
$ \bar{AB} =60, \bar{BC}=\bar{DA}= 36, \bar{CD}=108. $
$ M $ is the middle point of $ \bar {AB} $, and point $P$ on $ \bar{AM} $ follows that $ \bar {AP} $ =10.
The foot of perpendicular dropped from $P$ to $ \bar {BD} $ is $E$.
$ \bar{AC} \cap \bar{BD} $ is $ F $.
Point $X$ is on $ \bar {AF} $ which follows $ \bar{MX}=\bar{ME} $
Find $ \bar{AX} \times \bar{AF} $