Olympiad problems

2004 Bulgaria Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , reflection , rotation , trapezoid , power of a point

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

2007 Balkan MO, 1

Tags: trigonometry , geometry , geometric transformation , reflection , trapezoid , circumcircle , trig identities

Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.

2024 AMC 10, 13

Tags: geometric transformation , rotation , dilation

Two transformations are said to [i]commute[/i] if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane: - A translation $2$ units to the right - A $90^\circ$- rotation counterclockwise about the origin. - A reflection across the $x$-axis, and - A dilation centered at the origin with scale factor $2$. Of the $6$ pairs of distinct transformations from this list, how many commute? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5 \qquad $

2014 Harvard-MIT Mathematics Tournament, 10

Tags: geometry , circumcircle , geometric transformation , ratio , projective geometry

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $\Gamma$ be the circumcircle of $ABC$, let $O$ be its circumcenter, and let $M$ be the midpoint of minor arc $BC$. Circle $\omega_1$ is internally tangent to $\Gamma$ at $A$, and circle $\omega_2$, centered at $M$, is externally tangent to $\omega_1$ at a point $T$. Ray $AT$ meets segment $BC$ at point $S$, such that $BS - CS = \dfrac4{15}$. Find the radius of $\omega_2$

2004 Moldova Team Selection Test, 7

Tags: inequalities , geometry , circumcircle , inradius , function , trigonometry , geometric transformation

Let $ABC$ be a triangle, let $O$ be its circumcenter, and let $H$ be its orthocenter. Let $P$ be a point on the segment $OH$. Prove that $6r\leq PA+PB+PC\leq 3R$, where $r$ is the inradius and $R$ the circumradius of triangle $ABC$. [b]Moderator edit:[/b] This is true only if the point $P$ lies inside the triangle $ABC$. (Of course, this is always fulfilled if triangle $ABC$ is acute-angled, since in this case the segment $OH$ completely lies inside the triangle $ABC$; but if triangle $ABC$ is obtuse-angled, then the condition about $P$ lying inside the triangle $ABC$ is really necessary.)

1998 Turkey MO (2nd round), 2

Tags: geometric transformation , reflection , parallelogram , angle bisector , hjelmslev , geometry

Variable points $M$ and $N$ are considered on the arms $\left[ OX \right.$ and $\left[ OY \right.$ , respectively, of an angle $XOY$ so that $\left| OM \right|+\left| ON \right|$ is constant. Determine the locus of the midpoint of $\left[ MN \right]$.

2006 India IMO Training Camp, 1

Tags: circumcircle , geometric transformation , reflection , homothety , ratio , geometry

Let $ABC$ be a triangle and let $P$ be a point in the plane of $ABC$ that is inside the region of the angle $BAC$ but outside triangle $ABC$. [b](a)[/b] Prove that any two of the following statements imply the third. [list] [b](i)[/b] the circumcentre of triangle $PBC$ lies on the ray $\stackrel{\to}{PA}$. [b](ii)[/b] the circumcentre of triangle $CPA$ lies on the ray $\stackrel{\to}{PB}$. [b](iii)[/b] the circumcentre of triangle $APB$ lies on the ray $\stackrel{\to}{PC}$.[/list] [b](b)[/b] Prove that if the conditions in (a) hold, then the circumcentres of triangles $BPC,CPA$ and $APB$ lie on the circumcircle of triangle $ABC$.

2014 Contests, 2

Tags: geometric transformation , homothety , reflection , geometry

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]

2008 AIME Problems, 9

Tags: analytic geometry , geometry , geometric transformation , rotation , linear algebra , matrix

A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.

2002 Iran MO (3rd Round), 24

Tags: geometry , incenter , circumcircle , geometric transformation , homothety , ratio , trapezoid

$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$) Prove that $P$ and $I$ and $D$ are on a line.

1997 All-Russian Olympiad, 4

Tags: geometry , geometric transformation , pigeonhole principle , combinatorics

The numbers from $1$ to $100$ are arranged in a $10\times 10$ table so that any two adjacent numbers have sum no larger than $S$. Find the least value of $S$ for which this is possible. [i]D. Hramtsov[/i]

2008 Kyiv Mathematical Festival, 5

Tags: geometry , geometric transformation , rotation , search , combinatorics

Some $ m$ squares on the chessboard are marked. If among four squares at the intersection of some two rows and two columns three squares are marked then it is allowed to mark the fourth square. Find the smallest $ m$ for which it is possible to mark all squares after several such operations.

2019 BMT Spring, 6

Tags: geometry , geometric transformation , homothety

Let $ \triangle ABE $ be a triangle with $ \frac{AB}{3} = \frac{BE}{4} = \frac{EA}{5} $. Let $ D \neq A $ be on line $ \overline{AE} $ such that $ AE = ED $ and $ D $ is closer to $ E $ than to $ A $. Moreover, let $ C $ be a point such that $ BCDE $ is a parallelogram. Furthermore, let $ M $ be on line $ \overline{CD} $ such that $ \overline{AM} $ bisects $ \angle BAE $, and let $ P $ be the intersection of $ \overline{AM} $ and $ \overline{BE} $. Compute the ratio of $ PM $ to the perimeter of $ \triangle ABE $.

2015 India IMO Training Camp, 1

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.

1989 IMO Longlists, 63

Tags: geometric transformation , reflection , geometry

Let $ l_i,$ $ i \equal{} 1,2,3$ be three non-collinear straight lines in the plane, which build a triangle, and $ f_i$ the axial reflections in $ l_i$. Prove that for each point $ P$ in the plane there exists finite interconnections (compositions) of the reflections of $ f_i$ which carries $ P$ into the triangle built by the straight lines $ l_i,$ i.e. maps that point to a point interior to the triangle.

2012 AIME Problems, 13

Tags: geometric transformation , reflection , trigonometry , rotation , analytic geometry , 3d geometry , geometry

Three concentric circles have radii $3$, $4$, and $5$. An equilateral triangle with one vertex on each circle has side length $s$. The largest possible area of the triangle can be written as $a+\frac{b}{c}\sqrt{d}$, where $a,b,c$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d$.

2010 India IMO Training Camp, 7

Tags: geometry , circumcircle , geometric transformation , dilation , reflection , parallelogram , angle bisector

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.

1953 AMC 12/AHSME, 49

Tags: analytic geometry , geometry , geometric transformation , reflection

The coordinates of $ A,B$ and $ C$ are $ (5,5),(2,1)$ and $ (0,k)$ respectively. The value of $ k$ that makes $ \overline{AC}\plus{}\overline{BC}$ as small as possible is: $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4\frac{1}{2} \qquad\textbf{(C)}\ 3\frac{6}{7} \qquad\textbf{(D)}\ 4\frac{5}{6} \qquad\textbf{(E)}\ 2\frac{1}{7}$

2005 AMC 8, 21

Tags: geometry , geometric transformation , reflection , dilation

How many distinct triangles can be drawn using three of the dots below as vertices? [asy]dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));[/asy] $ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24 $

2006 China Team Selection Test, 2

Tags: geometry , circumcircle , geometric transformation , reflection , parallelogram , ratio , analytic geometry

Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$. Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.

2008 Iran Team Selection Test, 12

Tags: geometry , circumcircle , geometric transformation , reflection , symmetry , incenter , angle bisector

In the acute-angled triangle $ ABC$, $ D$ is the intersection of the altitude passing through $ A$ with $ BC$ and $ I_a$ is the excenter of the triangle with respect to $ A$. $ K$ is a point on the extension of $ AB$ from $ B$, for which $ \angle AKI_a\equal{}90^\circ\plus{}\frac 34\angle C$. $ I_aK$ intersects the extension of $ AD$ at $ L$. Prove that $ DI_a$ bisects the angle $ \angle AI_aB$ iff $ AL\equal{}2R$. ($ R$ is the circumradius of $ ABC$)

2003 Iran MO (3rd Round), 6

Tags: geometry , ratio , geometric transformation , homothety , circumcircle , projective geometry , cyclic quadrilateral

let the incircle of a triangle ABC touch BC,AC,AB at A1,B1,C1 respectively. M and N are the midpoints of AB1 and AC1 respectively. MN meets A1C1 at T . draw two tangents TP and TQ through T to incircle. PQ meets MN at L and B1C1 meets PQ at K . assume I is the center of the incircle . prove IK is parallel to AL

2012 Sharygin Geometry Olympiad, 8

Tags: geometry , incenter , trigonometry , geometric transformation , homothety

Let $BM$ be the median of right-angled triangle $ABC (\angle B = 90^{\circ})$. The incircle of triangle $ABM$ touches sides $AB, AM$ in points $A_{1},A_{2}$; points $C_{1}, C_{2}$ are defined similarly. Prove that lines $A_{1}A_{2}$ and $C_{1}C_{2}$ meet on the bisector of angle $ABC$.

2014 Bundeswettbewerb Mathematik, 2

Tags: geometry , 3d geometry , prism , algorithm , geometric transformation , reflection , induction

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

2014 Dutch IMO TST, 3

Tags: circumcircle , geometric transformation , reflection , trapezoid , geometry

Let $H$ be the orthocentre of an acute triangle $ABC$. The line through $A$ perpendicular to $AC$ and the line through $B$ perpendicular to $BC$ intersect in $D$. The circle with centre $C$ through $H$ intersects the circumcircle of triangle $ABC$ in the points $E$ and $F$. Prove that $|DE| = |DF| = |AB|$.