Olympiad problems

1986 IMO, 2

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.

1953 Moscow Mathematical Olympiad, 247

Tags: combinatorial geometry , combinatorics , triangle

Inside a convex $1000$-gon, $500$ points are selected so that no three of the $1500$ points — the ones selected and the vertices of the polygon — lie on the same straight line. This $1000$-gon is then divided into triangles so that all $1500$ points are vertices of the triangles, and so that these triangles have no other vertices. How many triangles will there be?

2012 Polish MO Finals, 3

Tags: circles , triangle , bisector , geometry

Triangle $ABC$ with $AB = AC$ is inscribed in circle $o$. Circles $o_1$ and $o_2$ are internally tangent to circle $o$ in points $P$ and $Q$, respectively, and they are tangent to segments $AB$ and $AC$, respectively, and they are disjoint with the interior of triangle $ABC$. Let $m$ be a line tangent to circles $o_1$ and $o_2$, such that points $P$ and $Q$ lie on the opposite side than point $A$. Line $m$ cuts segments $AB$ and $AC$ in points $K$ and $L$, respectively. Prove, that intersection point of lines $PK$ and $QL$ lies on bisector of angle $BAC$.

2003 Spain Mathematical Olympiad, Problem 3

Tags: triangle , geometry

The altitudes of the triangle ${ABC}$ meet in the point ${H}$. You know that ${AB = CH}$. Determine the value of the angle $\widehat{BCA}$.

1967 IMO Shortlist, 5

Tags: algebra , trigonometry , 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.

2012 Brazil Team Selection Test, 4

Tags: trigonometry , triangle inequality , algebra , triangle

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

2015 EGMO, 1

Tags: tangent , triangle , geometry

Let $\triangle ABC$ be an acute-angled triangle, and let $D$ be the foot of the altitude from $C.$ The angle bisector of $\angle ABC$ intersects $CD$ at $E$ and meets the circumcircle $\omega$ of triangle $\triangle ADE$ again at $F.$ If $\angle ADF = 45^{\circ}$, show that $CF$ is tangent to $\omega .$

2024 Brazil National Olympiad, 2

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

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.

1978 Bundeswettbewerb Mathematik, 1

Tags: algebra , triangle , inequalities

Let $a, b, c$ be sides of a triangle. Prove that $$\frac{1}{3} \leq \frac{a^2 +b^2 +c^2 }{(a+b+c)^2 } < \frac{1}{2}$$ and show that $\frac{1}{2}$ cannot be replaced with a smaller number.

1980 Bundeswettbewerb Mathematik, 2

Tags: triangle , bisector , distance , geometry

In a triangle $ABC$, the bisectors of angles $A$ and $B$ meet the opposite sides of the triangle at points $D$ and $E$, respectively. A point $P$ is arbitrarily chosen on the line $DE$. Prove that the distance of $P$ from line $AB$ equals the sum or the difference of the distances of $P$ from lines $AC$ and $BC$.

1997 Brazil Team Selection Test, Problem 1

Tags: triangle , geometry

In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.

2014 Bosnia and Herzegovina Junior BMO TST, 2

Tags: geometry , triangle , right angle

In triangle $ABC$, on line $CA$ it is given point $D$ such that $CD = 3 \cdot CA$ (point $A$ is between points $C$ and $D$), and on line $BC$ it is given point $E$ ($E \neq B$) such that $CE=BC$. If $BD=AE$, prove that $\angle BAC= 90^{\circ}$

2020 AIME Problems, 13

Tags: geometry , triangle

Point $D$ lies on side $BC$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC$. The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F$, respectively. Given that $AB=4$, $BC=5$, $CA=6$, the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt n}p$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

2005 Slovenia National Olympiad, Problem 3

Tags: triangle , geometry

In an isosceles triangle $ABC$ with $AB = AC$, $D$ is the midpoint of $AC$ and $E$ is the projection of $D$ onto $BC$. Let $F$ be the midpoint of $DE$. Prove that the lines $BF$ and $AE$ are perpendicular if and only if the triangle $ABC$ is equilateral.

1970 IMO, 3

Tags: counting , acute , triangle , point set , combinatorial geometry

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

2017 OMMock - Mexico National Olympiad Mock Exam, 2

Tags: game , triangle , board , combinatorics

Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have $k$ white tokens on a line. Determine the maximum value of $k$ for which Alice can achieve this no matter how Bob plays. [i]Proposed by Oriol Solé[/i]

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.

2020 Jozsef Wildt International Math Competition, W9

Tags: inequalities , geometry , triangle , inradius

In any triangle $ABC$ prove that the following relationship holds: $$\begin{vmatrix}(b+c)^2&a^2&a^2\\b^2&(c+a)^2&b^2\\c^2&c^2&(a+b)^2\end{vmatrix}\ge93312r^6$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

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 $\angle BAC$.

2018 Serbia National Math Olympiad, 1

Tags: geometry , incenter , moving points , triangle

Let $\triangle ABC$ be a triangle with incenter $I$. Points $P$ and $Q$ are chosen on segmets $BI$ and $CI$ such that $2\angle PAQ=\angle BAC$. If $D$ is the touch point of incircle and side $BC$ prove that $\angle PDQ=90$.

2018 EGMO, 5

Tags: power of a point , triangle , angle , geometry

Let $\Gamma $ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\Gamma$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle BCA$ intersects $\Omega$ at two different points $P$ and $Q$. Prove that $\angle ABP = \angle QBC$.

1998 IMO, 5

Tags: incenter , geometry , trigonometry , triangle

Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.

1982 Bundeswettbewerb Mathematik, 2

Tags: triangle , equilateral , projection , space , geometry

Decide whether every triangle $ABC$ in space can be orthogonally projected onto a plane such that the projection is an equilateral triangle $A'B'C'$.

1980 IMO Shortlist, 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 $BC = XY$.

2003 IMO Shortlist, 5

Tags: geometry , coordinate geometry , coordinates , triangle , circles , combinatorics

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]