Olympiad problems

2008 Harvard-MIT Mathematics Tournament, 10

Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.

2012 AMC 12/AHSME, 17

Tags: trigonometry , slope , geometric transformation , geometry , analytic geometry , graphing lines , dilation

Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $

1968 Czech and Slovak Olympiad III A, 3

Tags: geometry , geometric transformation , locus , reflection

Two segment $AB,CD$ of the same length are given in plane such that lines $AB,CD$ are not parallel. Consider a point $S$ with the following property: the image of segment $AB$ under point reflection with respect to $S$ is identical to the mirror-image of segment $CD$ with respect to some axis. Find the locus of all such points $S.$

2009 USA Team Selection Test, 4

Tags: parallelogram , geometry , ratio , reflection , trigonometry , similar triangles , geometric transformation

Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB \equal{} \angle CQB \equal{} 45^\circ$, $ \angle ABP \equal{} \angle QBC \equal{} 75^\circ$, $ \angle RAC \equal{} 105^\circ$, and $ RQ^2 \equal{} 6CM^2$, compute $ AC^2/AR^2$. [i]Zuming Feng.[/i]

2015 NIMO Problems, 6

Tags: trigonometry , geometric transformation , circumcircle , homothety , parallelogram , reflection , geometry

Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Sreejato Bhattacharya[/i]

2010 Sharygin Geometry Olympiad, 2

Tags: geometry , circumcircle , trigonometry , gauss , euler , homothety , geometric transformation

Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

2006 China Team Selection Test, 1

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

$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$). Prove that $l_1 || l_2$.

1991 AMC 12/AHSME, 24

Tags: geometric transformation , rotation , logarithm , geometry

The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$? $ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $

2011 ELMO Shortlist, 6

Tags: polynomial , algebra , geometric transformation , induction , counting , derangement , geometry

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

2022 CHMMC Winter (2022-23), 2

Tags: rotation , geometry , geometric transformation

Jonathan and Eric are standing one kilometer apart on a large, flat, empty field. Jonathan rotates an angle of $\theta = 120^o$ counterclockwise around Eric, then Eric moves half of the distance to Jonathan. They keep repeating the previous two movements in this order. After a very long time, their locations approach a point $P$ on the field. What is the distance, in kilometers, from Jonathan’s starting location to $P$?

2005 Romania National Olympiad, 3

Tags: rotation , geometry , trapezoid , reflection , homothety , geometric transformation

Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that a) $\triangle OMN \sim \triangle OBA$; b) $OE \perp AB$. [i]Claudiu-Stefan Popa[/i]

2014 IMS, 10

Tags: group theory , abstract algebra , vector , geometry , linear algebra , geometric transformation

Let $V$ be a $n-$dimensional vector space over a field $F$ with a basis $\{e_1,e_2, \cdots ,e_n\}$.Prove that for any $m-$dimensional linear subspace $W$ of $V$, the number of elements of the set $W \cap P$ is less than or equal to $2^m$ where $P=\{\lambda_1e_1 + \lambda_2e_2 + \cdots + \lambda_ne_n : \lambda_i=0,1\}$.

1985 Spain Mathematical Olympiad, 1

Tags: geometric transformation , geometry , transformation

Let $f : P\to P$ be a bijective map from a plane $P$ to itself such that: (i) $f (r)$ is a line for every line $r$, (ii) $f (r) $ is parallel to $r$ for every line $r$. What possible transformations can $f$ be?

1995 Italy TST, 2

Tags: symmetry , matrix , geometry , geometric transformation , rectangle , linear algebra , reflection

Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?

2009 CentroAmerican, 2

Tags: cyclic quadrilateral , geometry , geometric transformation , homothety , reflection , power of a point , radical axis

\item Two circles $ \Gamma_1$ and $ \Gamma_2$ intersect at points $ A$ and $ B$. Consider a circle $ \Gamma$ contained in $ \Gamma_1$ and $ \Gamma_2$, which is tangent to both of them at $ D$ and $ E$ respectively. Let $ C$ be one of the intersection points of line $ AB$ with $ \Gamma$, $ F$ be the intersection of line $ EC$ with $ \Gamma_2$ and $ G$ be the intersection of line $ DC$ with $ \Gamma_1$. Let $ H$ and $ I$ be the intersection points of line $ ED$ with $ \Gamma_1$ and $ \Gamma_2$ respectively. Prove that $ F$, $ G$, $ H$ and $ I$ are on the same circle.

2010 India National Olympiad, 5

Tags: geometric transformation , circumcircle , geometry , euler , perpendicular bisector , reflection

Let $ ABC$ be an acute-angled triangle with altitude $ AK$. Let $ H$ be its ortho-centre and $ O$ be its circum-centre. Suppose $ KOH$ is an acute-angled triangle and $ P$ its circum-centre. Let $ Q$ be the reflection of $ P$ in the line $ HO$. Show that $ Q$ lies on the line joining the mid-points of $ AB$ and $ AC$.

2022 Vietnam TST, 4

Tags: circumcircle , geometry , geometric transformation , reflection

An acute, non-isosceles triangle $ABC$ is inscribed in a circle with centre $O$. A line go through $O$ and midpoint $I$ of $BC$ intersects $AB, AC$ at $E, F$ respectively. Let $D, G$ be reflections to $A$ over $O$ and circumcentre of $(AEF)$, respectively. Let $K$ be the reflection of $O$ over circumcentre of $(OBC)$. $a)$ Prove that $D, G, K$ are collinear. $b)$ Let $M, N$ are points on $KB, KC$ that $IM\perp AC$, $IN\perp AB$. The midperpendiculars of $IK$ intersects $MN$ at $H$. Assume that $IH$ intersects $AB, AC$ at $P, Q$ respectively. Prove that the circumcircle of $\triangle APQ$ intersects $(O)$ the second time at a point on $AI$.

2005 Baltic Way, 11

Tags: circumcircle , geometry , geometric transformation , rotation , rhombus , power of a point , radical axis

Let the points $D$ and $E$ lie on the sides $BC$ and $AC$, respectively, of the triangle $ABC$, satisfying $BD=AE$. The line joining the circumcentres of the triangles $ADC$ and $BEC$ meets the lines $AC$ and $BC$ at $K$ and $L$, respectively. Prove that $KC=LC$.

2010 Romania Team Selection Test, 1

Tags: parallelogram , geometric transformation , rectangle , geometry , homothety

Let $P$ be a point in the plane and let $\gamma$ be a circle which does not contain $P$. Two distinct variable lines $\ell$ and $\ell'$ through $P$ meet the circle $\gamma$ at points $X$ and $Y$, and $X'$ and $Y'$, respectively. Let $M$ and $N$ be the antipodes of $P$ in the circles $PXX'$ and $PYY'$, respectively. Prove that the line $MN$ passes through a fixed point. [i]Mihai Chis[/i]

2006 China Team Selection Test, 1

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

$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$). Prove that $l_1 || l_2$.

2010 AIME Problems, 15

Tags: circumcircle , geometry , ratio , spiral similarity , angle bisector , geometric transformation

In triangle $ ABC$, $ AC \equal{} 13, BC \equal{} 14,$ and $ AB\equal{}15$. Points $ M$ and $ D$ lie on $ AC$ with $ AM\equal{}MC$ and $ \angle ABD \equal{} \angle DBC$. Points $ N$ and $ E$ lie on $ AB$ with $ AN\equal{}NB$ and $ \angle ACE \equal{} \angle ECB$. Let $ P$ be the point, other than $ A$, of intersection of the circumcircles of $ \triangle AMN$ and $ \triangle ADE$. Ray $ AP$ meets $ BC$ at $ Q$. The ratio $ \frac{BQ}{CQ}$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\minus{}n$.

1979 Spain Mathematical Olympiad, 2

Tags: geometric transformation , combinatorics

A certain Oxford professor, assigned to espionage cryptography services British, role played by Dirk Bogarde in a film, recruits his proposing small attention exercises, such as mentally reading a word the other way around. Frequently he does it with his own name: $SEBASTIAN$, what will there be to read $NAITSABES$. He wonders if there is any movement of the plane or of space that transforms one of these words in the other, just as they appear written. And if it had been called $AVITO$, like a certain Unamuno character? Give a reasoned explanation for each answer.

1984 Iran MO (2nd round), 3

Tags: function , algebra , geometry , calculus , derivative , geometric transformation

Let $f : \mathbb R \to \mathbb R$ be a function such that \[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\] Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$

2014 India IMO Training Camp, 1

Tags: geometry , trigonometry , geometric transformation , circumcircle , dilation , radical axis , reflection

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

2006 Korea Junior Math Olympiad, 4

Tags: analytic geometry , geometric transformation , algebra , transformation

In the coordinate plane, define $M = \{(a, b),a,b \in Z\}$. A transformation $S$, which is defined on $M$, sends $(a,b)$ to $(a + b, b)$. Transformation $T$, also defined on $M$, sends $(a, b)$ to $(-b, a)$. Prove that for all $(a, b) \in M$, we can use $S,T$ denitely to map it to $(g,0)$.