Olympiad problems

2004 Purple Comet Problems, 17

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

We want to paint some identically-sized cubes so that each face of each cube is painted a solid color and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce?

1985 Federal Competition For Advanced Students, P2, 3

Tags: geometry , geometric transformation , projective geometry , reflection

A line meets the lines containing sides $ BC,CA,AB$ of a triangle $ ABC$ at $ A_1,B_1,C_1,$ respectively. Points $ A_2,B_2,C_2$ are symmetric to $ A_1,B_1,C_1$ with respect to the midpoints of $ BC,CA,AB,$ respectively. Prove that $ A_2,B_2,$ and $ C_2$ are collinear.

1988 IMO Longlists, 40

Tags: rectangle , circumcircle , geometric transformation , symmetry , homothety , ratio , geometry

[b]i.)[/b] Consider a circle $K$ with diameter $AB;$ with circle $L$ tangent to $AB$ and to $K$ and with a circle $M$ tangent to circle $K,$ circle $L$ and $AB.$ Calculate the ration of the area of circle $K$ to the area of circle $M.$ [b]ii.)[/b] In triangle $ABC, AB = AC$ and $\angle CAB = 80^{\circ}.$ If points $D,E$ and $F$ lie on sides $BC, AC$ and $AB,$ respectively and $CE = CD$ and $BF = BD,$ then find the size of $\angle EDF.$

1948 Moscow Mathematical Olympiad, 152

Tags: geometric transformation , geometry , combinatorial geometry , angle , 3d geometry , reflection

a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections. b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

1998 Turkey MO (2nd round), 2

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

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 $\left[ MN \right]$.

2008 AIME Problems, 13

Tags: geometry , reflection , geometric transformation , integration , analytic geometry , trigonometry , calculus

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $ R$ be the region outside the hexagon, and let $ S\equal{}\{\frac{1}{z}|z\in R\}$. Then the area of $ S$ has the form $ a\pi\plus{}\sqrt{b}$, where $ a$ and $ b$ are positive integers. Find $ a\plus{}b$.

2013 Sharygin Geometry Olympiad, 10

Tags: geometric transformation , ratio , geometry , asymptote , homothety

The incircle of triangle $ABC$ touches the side $AB$ at point $C'$; the incircle of triangle $ACC'$ touches the sides $AB$ and $AC$ at points $C_1, B_1$; the incircle of triangle $BCC'$ touches the sides $AB$ and $BC$ at points $C_2$, $A_2$. Prove that the lines $B_1C_1$, $A_2C_2$, and $CC'$ concur.

KoMaL A Problems 2017/2018, A. 705

Tags: geometry , reflection , geometric transformation

Triangle $ABC$ has orthocenter $H$. Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$. Suppose that circle $BHD$ meets $AB$ at $P\ne B$, and circle $CHD$ meets $AC$ at $Q\ne C$. Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle. [i]Michael Ren[/i]

2005 AMC 8, 21

Tags: reflection , geometric transformation , geometry , 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 $

2014 Iran Team Selection Test, 6

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

$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively . Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$. Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$. Prove that the circumcenter of $AIT$ is on the $BC$.

2014 Iran MO (3rd Round), 5

Tags: rotation , combinatorics , geometric transformation , geometry

An $n$-mino is a connected figure made by connecting $n$ $1 \times 1 $ squares. Two polyminos are the same if moving the first we can reach the second. For a polymino $P$ ,let $|P|$ be the number of $1 \times 1$ squares in it and $\partial P$ be number of squares out of $P$ such that each of the squares have at least on edge in common with a square from $P$. (a) Prove that for every $x \in (0,1)$:\[\sum_P x^{|P|}(1-x)^{\partial P}=1\] The sum is on all different polyminos. (b) Prove that for every polymino $P$, $\partial P \leq 2|P|+2$ (c) Prove that the number of $n$-minos is less than $6.75^n$. [i]Proposed by Kasra Alishahi[/i]

2008 Poland - Second Round, 2

Tags: reflection , geometric transformation , geometry

In the convex pentagon $ ABCDE$ following equalities holds: $ \angle ABD\equal{} \angle ACE, \angle ACB\equal{}\angle ACD, \angle ADC\equal{}\angle ADE$ and $ \angle ADB\equal{}\angle AEC$. The point $S$ is the intersection of the segments $BD$ and $CE$. Prove that lines $AS$ and $CD$ are perpendicular.

2008 China Team Selection Test, 1

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.

2009 Princeton University Math Competition, 3

Tags: reflection , symmetry , geometric transformation , rotation , geometry

A polygon is called concave if it has at least one angle strictly greater than $180^{\circ}$. What is the maximum number of symmetries that an 11-sided concave polygon can have? (A [i]symmetry[/i] of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.)

2007 Romania Team Selection Test, 4

Tags: reflection , function , geometry , geometric transformation

Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.

2006 India IMO Training Camp, 1

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

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 $ABC$.

2012 Turkmenistan National Math Olympiad, 8

Tags: trigonometry , geometric transformation , geometry

Let $ABC$ be a triangle inscribed in a circle of radius $1$. If the triangle's sides are integer numbers, then find that triangle's sides.

2015 India IMO Training Camp, 1

Tags: reflection , geometric transformation , circumcircle , geometry

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.

2008 AIME Problems, 5

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

A right circular cone has base radius $ r$ and height $ h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $ 17$ complete rotations. The value of $ h/r$ can be written in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

2010 Today's Calculation Of Integral, 559

Tags: geometric transformation , rotation , trigonometry , integration , calculus , calculus computations , geometry

In $ xyz$ space, consider two points $ P(1,\ 0,\ 1),\ Q(\minus{}1,\ 1,\ 0).$ Let $ S$ be the surface generated by rotation the line segment $ PQ$ about $ x$ axis. Answer the following questions. (1) Find the volume of the solid bounded by the surface $ S$ and two planes $ x\equal{}1$ and $ x\equal{}\minus{}1$. (2) Find the cross-section of the solid in (1) by the plane $ y\equal{}0$ to sketch the figure on the palne $ y\equal{}0$. (3) Evaluate the definite integral $ \int_0^1 \sqrt{t^2\plus{}1}\ dt$ by substitution $ t\equal{}\frac{e^s\minus{}e^{\minus{}s}}{2}$. Then use this to find the area of (2).

1988 Vietnam National Olympiad, 3

Tags: reflection , geometric transformation , geometry

Let $ a$, $ b$, $ c$ be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if $ S_a \circ S_b \circ S_c$ is a reflection in a line, where $ S_x$ denotes the reflection in line $ x$.

2004 Iran MO (3rd Round), 2

Tags: geometric transformation , homothety , ratio , combinatorics , geometry

$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$, such that for every line $XX'$ passing through $O$, where $X$ and $X'$ are boundary points of $A$, then \[ \frac12 \leq \frac {OX}{OX'} \leq 2.\]

1995 Baltic Way, 14

Tags: geometric transformation , reflection , combinatorics , geometry

There are $n$ fleas on an infinite sheet of triangulated paper. Initially the fleas are in different small triangles, all of which are inside some equilateral triangle consisting of $n^2$ small triangles. Once a second each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. For which natural numbers $n$ does there exist an initial configuration such that after a finite number of jumps all the $n$ fleas can meet in a single small triangle?

1990 IMO Longlists, 74

Tags: parallelogram , analytic geometry , homothety , geometric transformation , vector , geometry , ratio

Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.

2005 Croatia National Olympiad, 2

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

Let $U$ be the incenter of a triangle $ABC$ and $O_{1}, O_{2}, O_{3}$ be the circumcenters of the triangles $BCU, CAU, ABU$ , respectively. Prove that the circumcircles of the triangles $ABC$ and $O_{1}O_{2}O_{3}$ have the same center.