Olympiad problems

1991 IMO, 2

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

1967 IMO Shortlist, 5

Tags: trigonometry , algebra , triangle , trigonometric identities

Show that a triangle whose angles $A$, $B$, $C$ satisfy the equality \[ \frac{\sin^2 A + \sin^2 B + \sin^2 C}{\cos^2 A + \cos^2 B + \cos^2 C} = 2 \] is a rectangular triangle.

1990 Bulgaria National Olympiad, Problem 5

Tags: geometry , arc , circles , triangle

Given a circular arc, find a triangle of the smallest possible area which covers the arc so that the endpoints of the arc lie on the same side of the triangle.

1979 Bulgaria National Olympiad, Problem 5

Tags: geometry , pentagon , triangle

A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$. Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$, respectively. Prove that the triangles $SME$ and $SND$ are similar.

2024 Brazil National Olympiad, 2

Tags: geometry , triangle , congruent triangles , circumcircle , parallel , cyclic quadrilateral , midpoint

Let $ ABC $ be a scalene triangle. Let $ E $ and $ F $ be the midpoints of sides $ AC $ and $ AB $, respectively, and let $ D $ be any point on segment $ BC $. The circumcircles of triangles $ BDF $ and $ CDE $ intersect line $ EF $ at points $ K \neq F $, and $ L \neq E $, respectively, and intersect at points $ X \neq D $. The point $ Y $ is on line $ DX $ such that $ AY $ is parallel to $ BC $. Prove that points $ K $, $ L $, $ X $, and $ Y $ lie on the same circle.

1987 Bulgaria National Olympiad, Problem 5

Tags: geometry , triangle

Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$.

2023 Romanian Master of Mathematics Shortlist, C1

Tags: combinatorial geometry , orientation , combinatorics , double counting , triangle

Determine all integers $n \geq 3$ for which there exists a conguration of $n$ points in the plane, no three collinear, that can be labelled $1$ through $n$ in two different ways, so that the following condition be satisfied: For every triple $(i,j,k), 1 \leq i < j < k \leq n$, the triangle $ijk$ in one labelling has the same orientation as the triangle labelled $ijk$ in the other, except for $(i,j,k) = (1,2,3)$.

2015 ISI Entrance Examination, 7

Tags: geometry , circles , triangle

Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$

1982 IMO Longlists, 36

Tags: geometry , triangle , midpoint , incircle , concurrency

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

2020 Iranian Geometry Olympiad, 2

Tags: rectangle , circumcircle , triangle , geometry , angle

Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$. [i]Proposed by Ali Zamani[/i]

1992 IMO Shortlist, 11

Tags: trigonometry , geometry , law of sines , triangle

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

1978 IMO Longlists, 41

Tags: geometry , inradius , circumcircle , incenter , triangle

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

1978 IMO Longlists, 19

Tags: geometry , triangle

We consider three distinct half-lines $Ox, Oy, Oz$ in a plane. Prove the existence and uniqueness of three points $A \in Ox, B \in Oy, C \in Oz$ such that the perimeters of the triangles $OAB,OBC,OCA$ are all equal to a given number $2p > 0.$

1988 IMO Longlists, 34

Tags: circumcircle , triangle , concurrency , geometry

Let $ ABC$ be an acute-angled triangle. The lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are constructed through the vertices $ A$, $ B$ and $ C$ respectively according the following prescription: Let $ H$ be the foot of the altitude drawn from the vertex $ A$ to the side $ BC$; let $ S_{A}$ be the circle with diameter $ AH$; let $ S_{A}$ meet the sides $ AB$ and $ AC$ at $ M$ and $ N$ respectively, where $ M$ and $ N$ are distinct from $ A$; then let $ L_{A}$ be the line through $ A$ perpendicular to $ MN$. The lines $ L_{B}$ and $ L_{C}$ are constructed similarly. Prove that the lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are concurrent.

1969 IMO Shortlist, 21

Tags: geometry , trapezoid , incenter , circumcircle , triangle

$(FRA 4)$ A right-angled triangle $OAB$ has its right angle at the point $B.$ An arbitrary circle with center on the line $OB$ is tangent to the line $OA.$ Let $AT$ be the tangent to the circle different from $OA$ ($T$ is the point of tangency). Prove that the median from $B$ of the triangle $OAB$ intersects $AT$ at a point $M$ such that $MB = MT.$

2013 Singapore Senior Math Olympiad, 1

Tags: geometry , circumcircle , altitude , isosceles triangle , right triangle , length , triangle

In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC

2000 Moldova National Olympiad, Problem 7

Tags: triangle , geometry

In an isosceles triangle $ABC$ with $BC=AC$, $I$ is the incenter and $O$ the circumcenter. The line through $I$ parallel to $AC$ meets $BC$ at $D$. Prove that the lines $DO$ and $BI$ are perpendicular.

2019 Jozsef Wildt International Math Competition, W. 38

Tags: inequalities , triangle

Let $a$, $b$, $c$ be the sides of an acute triangle $\triangle ABC$ , then for any $x, y, z \geq 0$, such that $xy+yz+zx=1$ holds inequality:$$a^2x + b^2y + c^2z \geq 4F$$ where $F$ is the area of the triangle $\triangle ABC$

2013 Brazil Team Selection Test, 3

Tags: geometry , circumcircle , trigonometry , triangle , reflection

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

1997 IMO Shortlist, 20

Tags: circumcircle , cyclic quadrilateral , triangle , geometry

A quick solution: Let R be the foot of the perpend. from X to BC. Let's assume Q and R are in the interior of the segms AC and BC (respectively) and P in the ext of AD. P, R, Q are colinear (Simson's thm). PQ tangent to circle XRD iff XRQ=XDR iff Pi-XCA=XDR iff XBA=XDR=XDC=ADB iff XBC+ABC=ADB=DAC+ACB iff XAC+ABC=DAC+ACD iff ABC=ACD=ACB iff AB=AC. It's the same for all the other cases.

2019 IMO Shortlist, G4

Tags: geometry , triangle

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

2007 IMO, 4

Tags: geometry , circumcircle , incenter , triangle

In triangle $ ABC$ the bisector of angle $ BCA$ intersects the circumcircle again at $ R$, the perpendicular bisector of $ BC$ at $ P$, and the perpendicular bisector of $ AC$ at $ Q$. The midpoint of $ BC$ is $ K$ and the midpoint of $ AC$ is $ L$. Prove that the triangles $ RPK$ and $ RQL$ have the same area. [i]Author: Marek Pechal, Czech Republic[/i]

2001 IMO, 1

Tags: geometry , circumcenter , triangle , geometric inequality

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

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 $IJ = AH$.

2002 India IMO Training Camp, 4

Tags: geometry , circumcircle , orthocenter , triangle , concurrency

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.