Olympiad problems

2001 Iran MO (2nd round), 2

Tags: geometry , incenter , trigonometry , circumcircle , perpendicular bisector , geometry proposed

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

2006 Iran MO (3rd Round), 2

Tags: geometry , circumcircle , parallelogram , rectangle , geometric transformation , geometry proposed

$ABC$ is a triangle and $R,Q,P$ are midpoints of $AB,AC,BC$. Line $AP$ intersects $RQ$ in $E$ and circumcircle of $ABC$ in $F$. $T,S$ are on $RP,PQ$ such that $ES\perp PQ,ET\perp RP$. $F'$ is on circumcircle of $ABC$ that $FF'$ is diameter. The point of intersection of $AF'$ and $BC$ is $E'$. $S',T'$ are on $AB,AC$ that $E'S'\perp AB,E'T'\perp AC$. Prove that $TS$ and $T'S'$ are perpendicular.

2009 Indonesia TST, 3

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

Let $ ABC$ be an acute triangle with $ \angle BAC\equal{}60^{\circ}$. Let $ P$ be a point in triangle $ ABC$ with $ \angle APB\equal{}\angle BPC\equal{}\angle CPA\equal{}120^{\circ}$. The foots of perpendicular from $ P$ to $ BC,CA,AB$ are $ X,Y,Z$, respectively. Let $ M$ be the midpoint of $ YZ$. a) Prove that $ \angle YXZ\equal{}60^{\circ}$ b) Prove that $ X,P,M$ are collinear.

2008 China Team Selection Test, 1

Tags: algorithm , induction , geometry proposed , geometry

Prove that in a plane, arbitrary $ n$ points can be overlapped by discs that the sum of all the diameters is less than $ n$, and the distances between arbitrary two are greater than $ 1$. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)

2014 Iran Team Selection Test, 1

Tags: circumcircle , geometry proposed

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ . let $AI,BI,CI$ meets $BC,AC,AB$ at $A_{2},B_{2},C_{2}$. let $A'$ is a point on $AI$ such that $A_{1}A'\perp B_{2}C_{2}$ .$B',C'$ respectively. prove that two triangle $A'B'C',A_{1}B_{1}C_{1}$ are equal.

2013 JBMO TST - Turkey, 7

Tags: geometry , circumcircle , geometry proposed

In a convex quadrilateral $ABCD$ diagonals intersect at $E$ and $BE = \sqrt{2}\cdot ED, \: \angle BEC = 45^{\circ}.$ Let $F$ be the foot of the perpendicular from $A$ to $BC$ and $P$ be the second intersection point of the circumcircle of triangle $BFD$ and line segment $DC$. Find $\angle APD$.

2013 ELMO Shortlist, 12

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

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]

2007 Pre-Preparation Course Examination, 1

Tags: inequalities , geometry , inradius , geometry proposed

$D$ is an arbitrary point inside triangle $ABC$, and $E$ is inside triangle $BDC$. Prove that \[\frac{S_{DBC}}{(P_{DBC})^{2}}\geq\frac{S_{EBC}}{(P_{EBC})^{2}}\]

2007 Bulgaria National Olympiad, 2

Tags: geometry , geometry proposed

Find the least real number $m$ such that with all $5$ equilaterial triangles with sum of areas $m$ we can cover an equilaterial triangle with side 1. [i]O. Mushkarov, N. Nikolov[/i]

2001 Macedonia National Olympiad, 3

Tags: geometry , projective geometry , analytic geometry , circumcircle , symmetry , trigonometry , geometry proposed

Let $ABC$ be a scalene triangle and $k$ be its circumcircle. Let $t_A,t_B,t_C$ be the tangents to $k$ at $A, B, C,$ respectively. Prove that points $AB\cap t_C$, $CA\cap t_B$, and $BC\cap t_A$ exist, and that they are collinear.

1997 Romania Team Selection Test, 1

Tags: vector , ratio , geometry proposed , geometry

Let $ABCDEF$ be a convex hexagon, and let $P= AB \cap CD$, $Q = CD \cap EF$, $R = EF \cap AB$, $S = BC \cap DE$, $T = DE \cap FA$, $U = FA \cap BC$. Prove that $\frac{PQ}{CD} = \frac{QR}{EF} = \frac{RP}{AB}$ if and only if $\frac{ST}{DE} = \frac{TU}{FA} = \frac{US}{BC}$

1977 IMO Longlists, 1

Tags: geometry , 3D geometry , pyramid , geometry proposed

A pentagon $ABCDE$ inscribed in a circle for which $BC<CD$ and $AB<DE$ is the base of a pyramid with vertex $S$. If $AS$ is the longest edge starting from $S$, prove that $BS>CS$.

2010 Slovenia National Olympiad, 2

Tags: geometry proposed , geometry

Let $\mathfrak K_1$ and $\mathfrak K_2$ be circles centered at $O_1$ and $O_2,$ respectively, meeting at the points $A$ and $B.$ Let $p$ be the line through the point $A$ meeting the circles $\mathfrak K_1$ and $\mathfrak K_2$ again at $C_1$ and $C_2.$ Assume that $A$ lies between $C_1$ and $C_2.$ Denote the intersection of the lines $C_1O_1$ and $C_2O_2$ by $D.$ Prove that the points $C_1, C_2, B$ and $D$ lie on the same circle.

2009 Sharygin Geometry Olympiad, 23

Tags: geometry , 3D geometry , prism , geometry proposed

Is it true that for each $ n$, the regular $ 2n$-gon is a projection of some polyhedron having not greater than $ n \plus{} 2$ faces?

2004 CentroAmerican, 3

Tags: geometry , parallelogram , geometry proposed

$ABC$ is a triangle, and $E$ and $F$ are points on the segments $BC$ and $CA$ respectively, such that $\frac{CE}{CB}+\frac{CF}{CA}=1$ and $\angle CEF=\angle CAB$. Suppose that $M$ is the midpoint of $EF$ and $G$ is the point of intersection between $CM$ and $AB$. Prove that triangle $FEG$ is similar to triangle $ABC$.

1990 Turkey Team Selection Test, 4

Tags: geometry , analytic geometry , linear algebra , matrix , rotation , geometry proposed

Let $ABCD$ be a convex quadrilateral such that \[\begin{array}{rl} E,F \in [AB],& AE = EF = FB \\ G,H \in [BC],& BG = GH = HC \\ K,L \in [CD],& CK = KL = LD \\ M,N \in [DA],& DM = MN = NA \end{array}\] Let \[[NG] \cap [LE] = \{P\}, [NG]\cap [KF] = \{Q\},\] \[{[}MH] \cap [KF] = \{R\}, [MH]\cap [LE]=\{S\}\] Prove that [list=a][*]$Area(ABCD) = 9 \cdot Area(PQRS)$ [*] $NP=PQ=QG$ [/list]

2010 ELMO Shortlist, 5

Tags: geometry , circumcircle , geometry proposed

Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three. [i]Carl Lian.[/i]

1983 Iran MO (2nd round), 4

Tags: geometry , 3D geometry , sphere , geometry proposed

The point $M$ moves such that the sum of squares of the lengths from $M$ to faces of a cube, is fixed. Find the locus of $M.$

2016 Iranian Geometry Olympiad, 1

Tags: geometry , geometry proposed

Let the circles $\omega$ and $\omega^ \prime$ intersect in $A$ and $B$. Tangent to circle$\omega$ at $A$ intersects$\omega^ \prime$ in $C$ and tangent to circle $\omega^ \prime$ at $A$ intersects $\omega$ in $D$. Suppose that $CD$ intersects$\omega$ and $\omega^ \prime$ in $E$ and $F$, respectively (assume that $E$ is between $F$ and $C$). The perpendicular to $AC$ from $E$ intersects $\omega^ \prime$ in point $P$ and perpendicular to $AD$ from $F$ intersects$\omega$ in point $Q$ (The points $A, P$ and $Q$ lie on the same side of the line $CD$). Prove that the points $A, P$ and $Q$ are collinear. Proposed by Mahdi Etesami Fard

2005 Taiwan TST Round 2, 2

Tags: trigonometry , geometry , analytic geometry , geometry proposed

In $\triangle ABC$, $AD$ is the bisector of $\angle A$, and $E$, $F$ are the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. $H$ is the intersection of $BE$ and $CF$, and $G$, $I$ are the feet of the perpendiculars from $D$ to $BE$ and $CF$, respectively. Prove that both $AFEH$ and $AEIH$ are cyclic quadrilaterals.

2008 Portugal MO, 5

Tags: geometry , geometry proposed

Let $ABC$ be a right-angled triangle in $A$ such that $AB<AC$. Let $M$ be the midpoint of $BC$ and let $D$ be the intersection of $AC$ with the perpendicular line to $BC$ which passes through $M$. Let $E$ be the intersection point of the parallel line to $AC$ which passes through $M$ with the perpendicular line to $BD$ which passes through $B$. Prove that triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC=60^{\circ}$.

2008 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry , circumcircle , symmetry , geometry proposed

Consider the acute-angled triangle $ ABC$, altitude $ AD$ and point $ E$ - intersection of $ BC$ with diameter from $ A$ of circumcircle. Let $ M,N$ be symmetric points of $ D$ with respect to the lines $ AC$ and $ AB$ respectively. Prove that $ \angle{EMC} \equal{} \angle{BNE}$.

2013 Saint Petersburg Mathematical Olympiad, 5

Tags: symmetry , geometry proposed , geometry

Given quadrilateral $ABCD$ with $AB=BC=CD$. Let $AC\cap BD=O$, $X,Y$ are symmetry points of $O$ respect to midpoints of $BC$, $AD$, and $Z$ is intersection point of lines, which perpendicular bisects of $AC$, $BD$. Prove that $X,Y,Z$ are collinear.

2006 Pre-Preparation Course Examination, 1

Tags: inequalities , geometry , perimeter , geometry proposed

Show that for a triangle we have \[ \max \{am_a,bm_b,cm_c\} \leq sR \] where $m_a$ denotes the length of median of side $BC$ and $s$ is half of the perimeter of the triangle.

2014 ELMO Shortlist, 9

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

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]