Olympiad problems

1999 IMO Shortlist, 8

Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.

1987 IMO Longlists, 22

Tags: geometry , circumcircle , triangle , minimization , circumcenter

Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. [i]Proposed by United Kingdom.[/i]

2007 Germany Team Selection Test, 1

Tags: incenter , triangle , midpoint , congruent triangles , geometry

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

1979 IMO Shortlist, 17

Tags: triangle , geometry , angle

Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that \[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\] Determine the angles of triangle $PQR.$

2002 India IMO Training Camp, 19

Tags: geometry , circumcircle , trigonometry , triangle

Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that \[ \angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB. \] Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum \[ \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}. \]

2016 Singapore MO Open, 1

Tags: inversion , similarity , geometry , triangle

Let $D$ be a point in the interior of $\triangle{ABC}$ such that $AB=ab$, $AC=ac$, $BC=bc$, $AD=ad$, $BD=bd$, $CD=cd$. Show that $\angle{ABD}+\angle{ACD}=60^{\circ}$. Source: 2016 Singapore Mathematical Olympiad (Open) Round 2, Problem 1

1975 IMO Shortlist, 8

Tags: geometry , trigonometry , triangle , angle

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

1961 IMO Shortlist, 4

Tags: ratio , inequalities , geometry , triangle , intersection

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$

2020 Malaysia IMONST 1, 18

Tags: altitude , triangle , geometry

In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest side has length $12$. What is the length of the altitude (height) of the triangle that is perpendicular to the longest side?

2000 IMO Shortlist, 5

Tags: diophantine equation , number theory , inradius , perimeter , triangle

Prove that there exist infinitely many positive integers $ n$ such that $ p \equal{} nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.

1986 IMO, 2

Tags: rotation , fixed point , equilateral triangle , triangle , geometry

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

2012 Bosnia Herzegovina Team Selection Test, 5

Tags: geometry , circumcircle , trigonometry , triangle

Given is a triangle $\triangle ABC$ and points $M$ and $K$ on lines $AB$ and $CB$ such that $AM=AC=CK$. Prove that the length of the radius of the circumcircle of triangle $\triangle BKM$ is equal to the lenght $OI$, where $O$ and $I$ are centers of the circumcircle and the incircle of $\triangle ABC$, respectively. Also prove that $OI\perp MK$.

1958 Czech and Slovak Olympiad III A, 2

Tags: construction , triangle , geometry

Construct a triangle $ABC$ given the magnitude of the angle $BCA$ and lengths of height $h_c$ and median $m_c$. Discuss conditions of solvability.

1972 IMO Longlists, 8

Tags: geometry , combinatorial geometry , combinatorics , point set , triangle

We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$

1980 IMO Longlists, 1

Tags: trigonometry , geometry , triangle , perpendicular bisector

Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.

1994 Chile National Olympiad, 2

Tags: geometry , rectangle , triangle

Show that it is possible to cut any triangle into several pieces, so that a rectangle is formed when they are joined together.

1980 IMO Longlists, 8

Tags: geometry , function , area , triangle

Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$. Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$.

2013 EGMO, 1

Tags: geometry , circumcircle , triangle

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.

2003 Czech And Slovak Olympiad III A, 4

Tags: construction , triangle , geometry

Let be given an obtuse angle $AKS$ in the plane. Construct a triangle $ABC$ such that $S$ is the midpoint of $BC$ and $K$ is the intersection point of $BC$ with the bisector of $\angle BAC$.

2020 Tuymaada Olympiad, 6

Tags: geometry , circumcircle , triangle

An isosceles triangle $ABC$ ($AB = BC$) is given. Circles $\omega_1$ and $\omega_2$ with centres $O_1$ and $O_2$ lie in the angle $ABC$ and touch the sides $AB$ and $CB$ at $A$ and $C$ respectively, and touch each other externally at point $X$. The side $AC$ meets the circles again at points $Y$ and $Z$. $O$ is the circumcenter of the triangle $XYZ$. Lines $O_2 O$ and $O_1 O$ intersect lines $AB$ and $BC$ at points $C_1$ and $A_1$ respectively. Prove that $B$ is the circumcentre of the triangle $A_1 OC_1$.

1990 Czech and Slovak Olympiad III A, 3

Tags: geometry , 3d geometry , tetrahedron , obtuse , triangle , plane

Let $ABCDEFGH$ be a cube. Consider a plane whose intersection with the tetrahedron $ABDE$ is a triangle with an obtuse angle $\varphi.$ Determine all $\varphi>\pi/2$ for which there is such a plane.

2020 Adygea Teachers' Geometry Olympiad, 3

Tags: 3d geometry , tetrahedron , triangle , geometry

Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?

2016 India IMO Training Camp, 1

Tags: geometry , triangle

Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.

2014 Canadian Mathematical Olympiad Qualification, 6

Tags: geometry , midpoint , triangle

Given a triangle $A, B, C, X$ is on side $AB$, $Y$ is on side $AC$, and $P$ and $Q$ are on side $BC$ such that $AX = AY , BX = BP$ and $CY = CQ$. Let $XP$ and $YQ$ intersect at $T$. Prove that $AT$ passes through the midpoint of $PQ$.

1966 Bulgaria National Olympiad, Problem 3

Tags: geometry , triangle

(a) In the plane of the triangle $ABC$, find a point with the following property: its symmetrical points with respect to the midpoints of the sides of the triangle lie on the circumscribed circle. (b) Construct the triangle $ABC$ if it is known the positions of the orthocenter $H$, midpoint of the side $AB$ and the midpoint of the segment joining the feet of the heights through vertices $A$ and $B$.