Found problems: 25757
2006 Tuymaada Olympiad, 2
Let $ABC$ be a triangle, $G$ it`s centroid, $H$ it`s orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$). It is known that $MG=R$, where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$.
[i]Proposed by F. Bakharev[/i]
2020 New Zealand MO, 4
Let $\Gamma_1$ and $\Gamma_2$ be circles internally tangent at point $A$, with $\Gamma_1$ inside $\Gamma_2$. Let $BC$ be a chord of $\Gamma_2$ which is tangent to $\Gamma_1$ at point $D$. Prove that line $AD$ is the angle bisector of $\angle BAC$.
1996 National High School Mathematics League, 10
Give two congruent regular triangular pyramids, stick their bottom surfaces together. Then ,it becomes a hexahedron with all dihedral angles equal. The length of the shortest edge of the hexahedron is $2$. Then, the furthest distance between two vertexes is________.
2023 Hong Kong Team Selection Test, Problem 2
Giiven $\Delta ABC$, $\angle CAB=75^{\circ}$ and $\angle ACB=45^{\circ}$. $BC$ is extended to $T$ so that $BC=CT$. Let $M$ be the midpoint of the segment $AT$. Find $\angle BMC$.
2019 Romania National Olympiad, 3
Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation
$$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$
represent the vertices of an equilateral triangle in the complex plane.
Kyiv City MO Seniors Round2 2010+ geometry, 2021.11.3.1
Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line$ AB$ is a common internal tangent, and the line $CD$ is a common external tangent to these circles, where $A, C \in k_1$ and $B, D \in k_2$. Knowing that $AB=12$ and $CD =16$, find the value of the product $r_1r_2$.
MathLinks Contest 1st, 3
Prove that in any acute triangle with sides $a, b, c$ circumscribed in a circle of radius $R$ the following inequality holds:
$$\frac{\sqrt2}{4} <\frac{Rp}{2aR + bc} <\frac{1}{2}$$ where $p$ represents the semi-perimeter of the triangle.
2006 Taiwan TST Round 1, 2
Let $P$ be a point on the plane. Three nonoverlapping equilateral triangles $PA_1A_2$, $PA_3A_4$, $PA_5A_6$ are constructed in a clockwise manner. The midpoints of $A_2A_3$, $A_4A_5$, $A_6A_1$ are $L$, $M$, $N$, respectively. Prove that triangle $LMN$ is equilateral.
2005 MOP Homework, 1
In convex hexagon $ ABCDEF$ all sides have equal length and
$ \angle A\plus{}\angle C\plus{}\angle E\equal{}\angle B\plus{}\angle D\plus{}\angle F$.
Prove that the diagonals $ AD,BE,CF$ are concurrent.
2012 Romanian Masters In Mathematics, 5
Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours.
[i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]
2010 Sharygin Geometry Olympiad, 8
Bisectrices $AA_1$ and $BB_1$ of triangle $ABC$ meet in $I$. Segments $A_1I$ and $B_1I$ are the bases of isosceles triangles with opposite vertices $A_2$ and $B_2$ lying on line $AB$. It is known that line $CI$ bisects segment $A_2B_2$. Is it true that triangle $ABC$ is isosceles?
2011 Postal Coaching, 2
Let $ABC$ be an acute triangle with $\angle BAC = 30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$ , respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$ , respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$ . Prove that $\angle BPC = 90^{\circ}$.
2019 AMC 12/AHSME, 3
Which one of the following rigid transformations (isometries) maps the line segment $\overline{AB}$ onto the line segment $\overline{A'B'}$ so that the image of $A(-2,1)$ is $A'(2,-1)$ and the image of $B(-1,4)$ is $B'(1,-4)?$
$\textbf{(A) } $ reflection in the $y$-axis
$\textbf{(B) } $ counterclockwise rotation around the origin by $90^{\circ}$
$\textbf{(C) } $ translation by 3 units to the right and 5 units down
$\textbf{(D) } $ reflection in the $x$-axis
$\textbf{(E) } $ clockwise rotation about the origin by $180^{\circ}$
1999 Tuymaada Olympiad, 1
In the triangle $ABC$ we have $\angle ABC=100^\circ$, $\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and $\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$.
[i]St.Petersburg folklore[/i]
2019 Romania National Olympiad, 3
In the regular hexagonal prism $ABCDEFA_1B_1C_1D_1E_1F_1$, We construct $, Q$, the projections of point $A$ on the lines $A_1B$ and $A_1C$ repsectilvely. We construct $R,S$, the projections of point $D_1$ on the lines $A_1D$ and $C_1D$ respectively.
a) Determine the measure of the angle between the planes $(AQP)$ and $(D_1RS)$.
b) Show that $\angle AQP = \angle D_1RS$.
2022 Kazakhstan National Olympiad, 5
Given a cyclic quadrilateral $ABCD$, let it's diagonals intersect at the point $O$. Take the midpoints of $AD$ and $BC$ as $M$ and $N$ respectively. Take a point $S$ on the arc $AB$ not containing $C$ or $D$ such that $$\angle SMA=\angle SNB$$ Prove that if the diagonals of the quadrilateral made from the lines $SM$, $SN$, $AB$, and $CD$ intersect at the point $T$, then $S$, $O$, and $T$ are collinear.
2004 Italy TST, 1
At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained?
$(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$
$(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$
$(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$
2018 NZMOC Camp Selection Problems, 8
Let $\lambda$ be a line and let $M, N$ be two points on $\lambda$. Circles $\alpha$ and $\beta$ centred at $A$ and $B$ respectively are both tangent to $\lambda$ at $M$, with $A$ and $B$ being on opposite sides of $\lambda$. Circles $\gamma$ and $\delta$ centred at $C$ and $D$ respectively are both tangent to $\lambda$ at $N$, with $C$ and $D$ being on opposite sides of $\lambda$. Moreover $A$ and $C$ are on the same side of $\lambda$. Prove that if there exists a circle tangent to all circles $\alpha, \beta, \gamma, \delta$ containing all of them in its interior, then the lines $AC, BD$ and $\lambda$ are either concurrent or parallel.
2016 Sharygin Geometry Olympiad, 7
From the altitudes of an acute-angled triangle, a triangle can be composed. Prove that a triangle can be composed from the bisectors of this triangle.
2016 India PRMO, 6
Suppose a circle $C$ of radius $\sqrt2$ touches the $Y$ -axis at the origin $(0, 0)$. A ray of light $L$, parallel to the $X$-axis, reflects on a point $P$ on the circumference of $C$, and after reflection, the reflected ray $L'$ becomes parallel to the $Y$ -axis. Find the distance between the ray $L$ and the $X$-axis.
2021 Saudi Arabia Training Tests, 21
Let $ABCD$ be a cyclic quadrilateral with $O$ is circumcenter and $AC$ meets $BD$ at $I$ Suppose that rays $DA,CD$ meet at $E$ and rays $BA,CD$ meet at $F$. The Gauss line of $ABCD$ meets $AB,BC,CD,DA$ at points $M,N,P,Q$ respectively. Prove that the circle of diameter $OI$ is tangent to two circles $(ENQ), (FMP)$
2013 AIME Problems, 12
Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
2021-IMOC qualification, G0
Let $D,E,F$ be the midpoints of $BC$ ,$CA$, $AB$ in $\vartriangle ABC$ such that $AD= 9$, $BE= 12$, $CF= 15$. Calculate the area of $\vartriangle ABC$
2014 Sharygin Geometry Olympiad, 20
A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$, then
a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$;
b) point $O$ lies on the perpendicular bisector to $PQ$.
2023 Rioplatense Mathematical Olympiad, 4
A set of points on the plane is [i]antiparallelogram [/i] if any four points of the set are [b]not[/b] vertices of a parallelogram. Prove that for any set of $2023$ points on the plane, [b]no[/b] three of them are collinears, there exists a subset of $17$ points, such that this subset is antiparallelogram.