Found problems: 25757
2002 Polish MO Finals, 2
There is given a triangle $ABC$ in a space. A sphere does not intersect the plane of $ABC$. There are $4$ points $K, L, M, P$ on the sphere such that $AK, BL, CM$ are tangent to the sphere and $\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}$. Show that the sphere touches the circumsphere of $ABCP$.
1977 Dutch Mathematical Olympiad, 4
There are an even number of points in a plane. No three of them lie on one straight line. Half of the points are red, the other half are blue. Prove that there exists a connecting line of a red and a blue point such that in each of the half-planes bounded by that line the number of red points is equal to the number of blue points.
2015 Switzerland - Final Round, 4
Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.
2006 IMO Shortlist, 2
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
2009 Tournament Of Towns, 4
A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side. Prove that the $2009$-gon with vertices at the images of these reflections also has area $S.$
[i](4 points)[/i]
1998 Swedish Mathematical Competition, 2
$ABC$ is a triangle. Show that $c \ge (a+b) \sin \frac{C}{2}$
2012 Uzbekistan National Olympiad, 5
Given points $A,B,C$ and $D$ lie a circle. $AC\cap BD=K$. $I_1, I_2,I_3$ and $I_4$ incenters of $ABK,BCK,CDK,DKA$. $M_1,M_2,M_3,M_4$ midpoints of arcs $AB,BC,CA,DA$ . Then prove that $M_1I_1,M_2I_2,M_3I_3,M_4I_4$ are concurrent.
2016 Singapore MO Open, 1
Let $D$ be a point in the interior of $\triangle{ABC}$ such that $AB=ab$, $AC=ac$, $BC=bc$, $AD=ad$, $BD=bd$, $CD=cd$. Show that $\angle{ABD}+\angle{ACD}=60^{\circ}$.
Source: 2016 Singapore Mathematical Olympiad (Open) Round 2, Problem 1
2006 National Olympiad First Round, 5
Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|AB|+|BD|=|AC|$ and $m(\widehat{BAD})=m(\widehat{DAC})=30^\circ$. What is $m(\widehat{ACB})$?
$
\textbf{(A)}\ 30^\circ
\qquad\textbf{(B)}\ 40^\circ
\qquad\textbf{(C)}\ 45^\circ
\qquad\textbf{(D)}\ 48^\circ
\qquad\textbf{(E)}\ 50^\circ
$
Indonesia MO Shortlist - geometry, g1.1
Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
1962 All-Soviet Union Olympiad, 6
Given the lengths $AB$ and $BC$ and the fact that the medians to those two sides are perpendicular, construct the triangle $ABC$.
1975 Dutch Mathematical Olympiad, 4
Given is a rectangular plane coordinate system.
(a) Prove that it is impossible to find an equilateral triangle whose vertices have integer coordinates.
(b) In the plane the vertices $A, B$ and $C$ lie with integer coordinates in such a way that $AB = AC$. Prove that $\frac{d(A,BC)}{BC}$ is rational.
2014 Turkey Junior National Olympiad, 4
$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.
2009 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle and $A_1$ the foot of the internal bisector of angle $BAC$. Consider $d_A$ the perpendicular line from $A_1$ on $BC$. Define analogously the lines $d_B$ and $d_C$. Prove that lines $d_A, d_B$ and $d_C$ are concurrent if and only if triangle $ABC$ is isosceles.
2010 Junior Balkan Team Selection Tests - Romania, 4
Let a triangle $ABC$ , $O$ it's circumcenter , $H$ ortocenter and $M$ the midpoint of $AH$. The perpendicular at $M$ to line $OM$ meets $AB$ and $AC$ at points $P$, respective $Q$. Prove that $MP=MQ$.
Babis
2005 Singapore Senior Math Olympiad, 2
Consider the nonconvex quadrilateral $ABCD$ with $\angle C>180$ degrees. Let the side $DC$ extended to meet $AB$ at $F$ and the side $BC$ extended to meet $AD$ at $E$. A line intersects the interiors of the sides $AB,AD,BC,CD$ at points $K,L,J,I$ respectively. Prove that if $DI=CF$ and $BJ=CE$, then $KJ=IL$
2025 Bulgarian Winter Tournament, 12.2
In the plane are fixed two internally tangent circles $\omega$ and $\Omega$, so that $\omega$ is inside $\Omega$. Denote their common point by $T$. The point $A \neq T$ moves on $\Omega$ and point $B$ on $\Omega$ is such that $AB$ is tangent to $\omega$. The line through $B$, perpendicular to $AB$, meets the external angle bisector of $\angle ATB$ at $P$. Prove that, as $A$ varies on $\Omega$, the line $AP$ passes through a fixed point.
2017 NIMO Summer Contest, 11
Let $a, b, c, p, q, r > 0$ such that $(a,b,c)$ is a geometric progression and $(p, q, r)$ is an arithmetic progression. If \[a^p b^q c^r = 6 \quad \text{and} \quad a^q b^r c^p = 29\] then compute $\lfloor a^r b^p c^q \rfloor$.
[i]Proposed by Michael Tang[/i]
2022 Indonesia MO, 3
Let $ABCD$ be a rectangle. Points $E$ and $F$ are on diagonal $AC$ such that $F$ lies between $A$ and $E$; and $E$ lies between $C$ and $F$. The circumcircle of triangle $BEF$ intersects $AB$ and $BC$ at $G$ and $H$ respectively, and the circumcircle of triangle $DEF$ intersects $AD$ and $CD$ at $I$ and $J$ respectively. Prove that the lines $GJ, IH$ and $AC$ concur at a point.
2014 Belarus Team Selection Test, 1
Let $\Gamma_B$ and $\Gamma_C$ be excircles of an acute-angled triangle $ABC$ opposite to its vertices $B$ and $C$, respectively. Let $C_1$ and $L$ be the tangent points of $\Gamma_C$ and the side $AB$ and the line $BC$ respectively. Let $B_1$ and $M$ be the tangent points of $\Gamma_B$ and the side $AC$ and the line $BC$, respectively. Let $X$ be the point of intersection of the lines $LC_1$ and $MB_1$. Prove that $AX$ is equal to the inradius of the triangle $ABC$.
(A. Voidelevich)
2014 Brazil Team Selection Test, 3
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
2024 Austrian MO National Competition, 2
Let $ABC$ be an acute triangle with $AB>AC$. Let $D,E,F$ denote the feet of its altitudes on $BC,AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$. Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$.
[i](Karl Czakler)[/i]
2022 VN Math Olympiad For High School Students, Problem 7
Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point.
Let $H_a, H_b, H_c$ be the orthocenter of triangles $TBC, TCA, TAB$, respectively.
a) Prove that: $T$ is the centroid of the $\triangle H_aH_bH_c$.
b) Denote $D, E, F$ respectively by the intersections of $H_cH_b$ and the segment $BC$, $H_cH_a$ and the segment $CA$, $H_aH_b$ and the segment $AB$. Prove that: the triangle $DEF$ is equilateral.
c) Prove that: the lines passing through $D, E, F$ and are respectively perpendicular to $BC, CA, AB$ are concurrent at a point. Let that point be $S$.
d) Prove that: $TS$ is parallel to the [i]Euler[/i] line of the triangle $ABC$.
2003 Argentina National Olympiad, 4
The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.
2007 District Olympiad, 1
Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13 BC$, find the measures of the angles of the triangle $\vartriangle ABC$.