Olympiad problems

2000 USAMO, 2

Let $S$ be the set of all triangles $ABC$ for which \[ 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, \] where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.

2004 Olympic Revenge, 3

Tags: parallelogram , geometry , incircle , concurrency

$ABC$ is a triangle and $\omega$ its incircle. Let $P,Q,R$ be the intersections with $\omega$ and the sides $BC,CA,AB$ respectively. $AP$ cuts $\omega$ in $P$ and $X$. $BX,CX$ cut $\omega$ in $M,N$ respectively. Show that $MR,NQ,AP$ are parallel or concurrent.

2006 Kyiv Mathematical Festival, 2

Tags: perimeter , geometry

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] 2006 equilateral triangles are located in the square with side 1. The sum of their perimeters is equal to 300. Prove that at least three of them have a common point.

III Soros Olympiad 1996 - 97 (Russia), 11.2

Tags: rotation , geometry , geometric transformation

It is known that the graph of the function $y = f(x)$ after a rotation of $45^o$ around a certain point turns into the graph of the function $y = x^3 + ax^2 + 19x + 97$. At what $a$ is this possible?

2019 AIME Problems, 15

Tags: radical axis , symmedian , geometry

Let $\overline{AB}$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $\overline{AB}$. Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega$. Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega$. Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q$. Line $PQ$ intersects $\omega$ at $X$ and $Y$. Assume that $AP=5$, $PB=3$, $XY=11$, and $PQ^2 = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2008 Germany Team Selection Test, 2

Tags: trapezoid , geometry , circumcircle

Let $ ABCD$ be an isosceles trapezium with $ AB \parallel{} CD$ and $ \bar{BC} \equal{} \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.

2003 National Olympiad First Round, 25

Tags: trigonometry , perpendicular bisector , circumcircle , geometry

Let $ABC$ be an acute triangle and $O$ be its circumcenter. Let $D$ be the midpoint of $[AB]$. The circumcircle of $\triangle ADO$ meets $[AC]$ at $A$ and $E$. If $|AE|=7$, $|DE|=8$, and $m(\widehat{AOD}) = 45^\circ$, what is the area of $\triangle ABC$? $ \textbf{(A)}\ 56\sqrt 3 \qquad\textbf{(B)}\ 56 \sqrt 2 \qquad\textbf{(C)}\ 50 \sqrt 2 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ \text{None of the preceding} $

2018 Sharygin Geometry Olympiad, 5

Tags: arc midpoint , radical axis , geometry

Let $ABCD$ be a cyclic quadrilateral, $BL$ and $CN$ be the internal angle bisectors in triangles $ABD$ and $ACD$ respectively. The circumcircles of triangles $ABL$ and $CDN$ meet at points $P$ and $Q$. Prove that the line $PQ$ passes through the midpoint of the arc $AD$ not containing $B$.

1989 National High School Mathematics League, 9

Tags: function , geometry

Functions $f_0(x)=|x|,f_1(x)=|f_0(x)-1|,f_2(x)=|f_1(x)-2|$. Area of the closed part between the figure of $f_2(x)$ and $x$-axis is________.

2015 Sharygin Geometry Olympiad, 1

Tags: geometry , midpoint , trapezoid

In trapezoid $ABCD$ angles $A$ and $B$ are right, $AB = AD, CD = BC + AD, BC < AD$. Prove that $\angle ADC = 2\angle ABE$, where $E$ is the midpoint of segment $AD$. (V. Yasinsky)

1993 Chile National Olympiad, 2

Tags: rectangle , geometry , area

Given a rectangle, circumscribe a rectangle of maximum area.

IV Soros Olympiad 1997 - 98 (Russia), 9.6

Tags: combinatorial geometry , geometry , combinatorics

Cut an acute triangle, one of whose sides is equal to the altitude drawn, by two straight cuts, into four parts, from which you can fold a square.

2012 ELMO Shortlist, 7

Tags: inversion , geometry , circumcircle , trigonometry

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$. [i]Alex Zhu.[/i]

2003 Chile National Olympiad, 6

Tags: tangent , circumcircle , geometry

Consider a triangle $ ABC $. On the line $ AC $ take a point $ B_1 $ such that $ AB = AB_1 $ and in addition, $ B_1 $ and $ C $ are located on the same side of the line with respect to the point $ A $. The bisector of the angle $ A $ intersects the side $ BC $ at a point that we will denote as $ A_1 $. Let $ P $ and $ R $ be the circumscribed circles of the triangles $ ABC $ and $ A_1B_1C $ respectively. They intersect at points $ C $ and $ Q $. Prove that the tangent to the circle $ R $ at the point $ Q $ is parallel to the line $ AC $.

1993 IberoAmerican, 2

Tags: perpendicular bisector , combinatorics , circumcircle , rhombus , pigeonhole principle , geometry

Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$. (ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$. Find the number of points that $S$ may contain.

2005 All-Russian Olympiad Regional Round, 11.4

Tags: radical axis , power of a point , geometry , circumcircle

11.4 Let $AA_1$ and $BB_1$ are altitudes of an acute non-isosceles triangle $ABC$, $A'$ is a midpoint of $BC$ and $B'$ is a midpoint of $AC$. A segement $A_1B_1$ intersects $A'B'$ at point $C'$. Prove that $CC'\perp HO$, where $H$ is a orthocenter and $O$ is a circumcenter of $ABC$. ([i]L. Emel'yanov[/i])

2019 Grand Duchy of Lithuania, 3

Tags: right angle , perpendicular , geometry , circumcircle , orthocenter

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. The perpendicular bisector of segment $CH$ intersects the sides $AC$ and $BC$ in points $X$ and $Y$ , respectively. The lines $XO$ and $YO$ intersect the side $AB$ in points $P$ and $Q$, respectively. Prove that if $XP + Y Q = AB + XY$ then $\angle OHC = 90^o$.

2002 AIME Problems, 13

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

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ 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$.

2014 France Team Selection Test, 2

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

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

1971 Polish MO Finals, 2

Tags: reflection , geometric transformation , geometry

A pool table has the shape of a triangle whose angles are in a rational ratio. A ball positioned at an interior point of the table is hit by a stick. The ball reflects from the sides of the triangle according to the law of reflection. Prove that the ball will move only along a finite number of segments. (It is assumed that the ball does not reach the vertices of the triangle.)

2012 Kosovo National Mathematical Olympiad, 4

Tags: geometry

Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$. Find $\angle APB$.

1993 Swedish Mathematical Competition, 5

Tags: sequence , geometry

A triangle with sides $a,b,c$ and perimeter $2p$ is given. Is possible, a new triangle with sides $p-a$, $p-b$, $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?

2008 iTest Tournament of Champions, 5

Tags: 3d geometry , geometry

Two squares of side length $2$ are glued together along their boundary so that the four vertices of the first square are glued to the midpoints of the four sides of the other square, and vice versa. This gluing results in a convex polyhedron. If the square of the volume of this polyhedron is written in simplest form as $\tfrac{a+b\sqrt c}d$, what is the value of $a+b+c+d$?

2015 Azerbaijan JBMO TST, 3

Tags: geometry , circumcircle

Let $ABC$ be a triangle such that $AB$ is not equal to $AC$. Let $M$ be the midpoint of $BC$ and $H$ be the orthocenter of triangle $ABC$. Let $D$ be the midpoint of $AH$ and $O$ the circumcentre of triangle $BCH$. Prove that $DAMO$ is a parallelogram.

2018 ASDAN Math Tournament, 4

Tags: geometry

Let $AB$ be the diameter of a circle with center $O$ and radius $5$. Extend $AB$ past $A$ to a point $C$ such that $BC = 18$, and let $D$ be a point on the circle such that $CD$ lies tangent to the circle. Next, draw $E$ on $CD$ such that $OE \parallel BD$. Compute $DE$.