Olympiad problems

2008 Spain Mathematical Olympiad, 2

Given a circle, two fixed points $A$ and $B$ and a variable point $P$, all of them on the circle, and a line $r$, $PA$ and $PB$ intersect $r$ at $C$ and $D$, respectively. Find two fixed points on $r$, $M$ and $N$, such that $CM\cdot DN$ is constant for all $P$.

2000 Putnam, 3

Tags: geometry , rectangle , analytic geometry , quadratic , geometric transformation , reflection

The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area $5$, and the polygon $P_2P_4P_6P_8$ is a rectangle of area $4$, find the maximum possible area of the octagon.

2018 PUMaC Combinatorics A, 5

Tags: combinatorics , geometry , geometric transformation , rotation

How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

1991 Romania Team Selection Test, 5

Tags: incenter , circumcircle , parallelogram , geometric transformation , reflection , geometry

In a triangle $A_1A_2A_3$, the excribed circles corresponding to sides $A_2A_3$, $A_3A_1$, $A_1A_2$ touch these sides at $T_1$, $T_2$, $T_3$, respectively. If $H_1$, $H_2$, $H_3$ are the orthocenters of triangles $A_1T_2T_3$, $A_2T_3T_1$, $A_3T_1T_2$, respectively, prove that lines $H_1T_1$, $H_2T_2$, $H_3T_3$ are concurrent.

2004 Iran MO (3rd Round), 16

Tags: parallelogram , rectangle , geometric transformation , homothety , circumcircle , cyclic quadrilateral , geometry

Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .

2019 Czech-Polish-Slovak Junior Match, 5

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

Let $A_1A_2 ...A_{360}$ be a regular $360$-gon with centre $S$. For each of the triangles $A_1A_{50}A_{68}$ and $A_1A_{50}A_{69}$ determine, whether its images under some $120$ rotations with centre $S$ can have (as triangles) all the $360$ points $A_1, A_2, ..., A_{360}$ as vertices.

2014 Baltic Way, 12

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

Triangle $ABC$ is given. Let $M$ be the midpoint of the segment $AB$ and $T$ be the midpoint of the arc $BC$ not containing $A$ of the circumcircle of $ABC.$ The point $K$ inside the triangle $ABC$ is such that $MATK$ is an isosceles trapezoid with $AT\parallel MK.$ Show that $AK = KC.$

1992 India National Olympiad, 5

Tags: geometry , geometric transformation , power of a point

Two circles $C_1$ and $C_2$ intersect at two distinct points $P, Q$ in a plane. Let a line passing through $P$ meet the circles $C_1$ and $C_2$ in $A$ and $B$ respectively. Let $Y$ be the midpoint of $AB$ and let $QY$ meet the cirlces $C_1$ and $C_2$ in $X$ and $Z$ respectively. Show that $Y$ is also the midpoint of $XZ$.

2020 Vietnam National Olympiad, 4

Tags: geometry , geometric transformation , reflection , parallelogram

Let a non-isosceles acute triangle ABC with the circumscribed cycle (O) and the orthocenter H. D, E, F are the reflection of O in the lines BC, CA and AB. a) $H_a$ is the reflection of H in BC, A' is the reflection of A at O and $O_a$ is the center of (BOC). Prove that $H_aD$ and OA' intersect on (O). b) Let X is a point satisfy AXDA' is a parallelogram. Prove that (AHX), (ABF), (ACE) have a comom point different than A

2014 NIMO Problems, 3

Tags: geometry , geometric transformation , reflection

In triangle $ABC$, we have $AB=AC=20$ and $BC=14$. Consider points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. If the minimum value of the sum $BN + MN + MC$ is $x$, compute $100x$. [i]Proposed by Lewis Chen[/i]

2014 IberoAmerican, 2

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

Let $ABC$ be an acute triangle and $H$ its orthocenter. Let $D$ be the intersection of the altitude from $A$ to $BC$. Let $M$ and $N$ be the midpoints of $BH$ and $CH$, respectively. Let the lines $DM$ and $DN$ intersect $AB$ and $AC$ at points $X$ and $Y$ respectively. If $P$ is the intersection of $XY$ with $BH$ and $Q$ the intersection of $XY$ with $CH$, show that $H, P, D, Q$ lie on a circumference.

1967 IMO Longlists, 54

Tags: 3d geometry , rotation , combinatorics , geometric transformation , geometry

Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?

2013 USA TSTST, 3

Tags: analytic geometry , graphing lines , slope , geometric transformation , reflection , floor function

Divide the plane into an infinite square grid by drawing all the lines $x=m$ and $y=n$ for $m,n \in \mathbb Z$. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly $1/4$ of the squares are black and no two black squares are adjacent). Let $r$ and $s$ be odd integers, and let $(x,y)$ be a point in the interior of any white square such that $rx-sy$ is irrational. Shoot a laser out of this point with slope $r/s$; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.

1999 Korea Junior Math Olympiad, 4

Tags: geometry , geometric transformation , rotation

$C$ is the unit circle in some plane. $R$ is a square with side $a$. $C$ is fixed and $R$ moves(without rotation) on the plane, in such a way that its center stays inside $C$(including boundaries). Find the maximum value of the area drawn by the trace of $R$.

2008 AMC 10, 24

Tags: geometry , rhombus , trigonometry , geometric transformation , parallelogram

Quadrilateral $ABCD$ has $AB=BC=CD$, $\angle ABC=70^\circ$, and $\angle BCD=170^\circ$. What is the degree measure of $\angle BAD$? $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 80\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ 95$

2000 Harvard-MIT Mathematics Tournament, 3

Tags: geometry , 3d geometry , geometric transformation , rotation

Using $3$ colors, red, blue and yellow, how many different ways can you color a cube (modulo rigid rotations)?

2014 ELMO Shortlist, 13

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

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

2000 Polish MO Finals, 2

Tags: geometry , trigonometry , circumcircle , analytic geometry , geometric transformation , reflection , cyclic quadrilateral

Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.

2010 Serbia National Math Olympiad, 2

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

In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$. [i]Proposed by Marko Djikic[/i]

2013 Moldova Team Selection Test, 3

Tags: geometry , geometric transformation , reflection , circumcircle , ratio , angle bisector

Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.

1996 USAMO, 3

Tags: geometry , geometric transformation , reflection , trapezoid , inequalities , ratio , angle bisector

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

2007 Iran MO (3rd Round), 1

Tags: geometric transformation , reflection , parallelogram , geometry

Let $ ABC$, $ l$ and $ P$ be arbitrary triangle, line and point. $ A',B',C'$ are reflections of $ A,B,C$ in point $ P$. $ A''$ is a point on $ B'C'$ such that $ AA''\parallel l$. $ B'',C''$ are defined similarly. Prove that $ A'',B'',C''$ are collinear.

1992 AMC 12/AHSME, 12

Tags: geometry , geometric transformation , reflection

Let $y = mx + b$ be the image when the line $x - 3y + 11 = 0$ is reflected across the x-axis. The value of $m + b$ is $ \textbf{(A)}\ -6\qquad\textbf{(B)}\ -5\qquad\textbf{(C)}\ -4\qquad\textbf{(D)}\ -3\qquad\textbf{(E)}\ -2 $

1985 AIME Problems, 11

Tags: ellipse , analytic geometry , geometry , geometric transformation , reflection , conic

An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?

2003 India IMO Training Camp, 5

Tags: geometry , geometric transformation , reflection , combinatorics

On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.