Olympiad problems

2003 German National Olympiad, 2

Tags: geometry , incenter , circumcircle , midpoint , Center , Triangle

There are four circles $k_1 , k_2 , k_3$ and $k_4$ of equal radius inside the triangle $ABC$. The circle $k_1$ touches the sides $AB, CA$ and the circle $k_4 $, $k_2$ touches the sides $AB,BC$ and $k_4$, and $k_3$ touches the sides $AC, BC$ and $k_4.$ Prove that the center of $k_4$ lies on the line connecting the incenter and circumcenter of $ABC.$

1975 IMO, 3

Tags: geometry , trigonometry , Triangle , angles , IMO , IMO 1975

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] $QR = RP.$

1967 IMO Longlists, 11

Tags: counting , triangle inequality , combinatorics , Triangle , IMO Shortlist , IMO Longlist

Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$

2014 Czech-Polish-Slovak Match, 1

Tags: Trigonometric Equations , polynomial equation , Triangle , geometry

Prove that if the positive real numbers $a, b, c$ satisfy the equation \[a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2),\] then there is a triangle $ABC$ with internal angles $\alpha, \beta, \gamma$ such that \[\sin \alpha = a, \qquad \sin \beta = b, \qquad \sin \gamma= c.\]

2012 Bosnia and Herzegovina Junior BMO TST, 4

Tags: Triangle , perimeter , Inequality , geometry , inequalities

If $a$, $b$ and $c$ are sides of triangle which perimeter equals $1$, prove that: $a^2+b^2+c^2+4abc<\frac{1}{2}$

1994 French Mathematical Olympiad, Problem 4

Tags: geometry , circumcircle , Triangle , minimization , Circumcenter , IMO Shortlist

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]

1982 IMO, 2

Tags: geometry , Triangle , midpoint , incircle , concurrency , IMO , IMO 1982

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.

2018 Bangladesh Mathematical Olympiad, 2

Tags: geometry , circles , Triangle , contests

BdMO National 2018 Higher Secondary P2 $AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$

1974 IMO, 2

Tags: trigonometry , circumcircle , geometry , Triangle , Trigonometric inequality , IMO , IMO 1974

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

2016 Brazil Team Selection Test, 1

Tags: geometry , IMO Shortlist , 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 $IJ = AH$.

2008 Chile National Olympiad, 2

Tags: right triangle , isosceles , Triangle , geometry

Let $ABC$ be right isosceles triangle with right angle in $A$. Given a point $P$ inside the triangle, denote by $a, b$ and $c$ the lengths of $PA, PB$ and $PC$, respectively. Prove that there is a triangle whose sides have a length of $a\sqrt2 , b$ and $c$.

2024 Brazil National Olympiad, 2

Tags: geometry , Triangle , midpoints , congruent triangles , circumcircle , 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.

1993 Bundeswettbewerb Mathematik, 3

Tags: geometry , perpendicular bisector , angle bisector , Triangle

In the triangle $ABC$, let $A'$ be the intersection of the perpendicular bisector of $AB$ and the angle bisector of $\angle BAC$ and define $B', C'$ analogously. Prove that a) The triangle $ABC$ is equilateral if and only if $A' =B'.$ b) If $A', B'$ and $C'$ are distinct, we have $\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.$

2001 IMO Shortlist, 2

Tags: geometry , Circumcenter , Triangle , IMO , IMO 2001 , hojoo lee , 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 $\angle A+\angle COP < 90^{\circ}$.

2001 Grosman Memorial Mathematical Olympiad, 4

Tags: Triangle , min , perpendicular , distance , geometry

The lengths of the sides of triangle $ABC$ are $4,5,6$. For any point $D$ on one of the sides, draw the perpendiculars $DP, DQ$ on the other two sides. What is the minimum value of $PQ$?

1955 Moscow Mathematical Olympiad, 310

Tags: inequalities , sidelengths of triangle , Triangle

Let the inequality $$Aa(Bb + Cc) + Bb(Aa + Cc) + Cc(Aa + Bb) > \frac{ABc^2 + BCa^2 + CAb^2}{2}$$ with given $a > 0, b > 0, c > 0$ hold for all $A > 0, B > 0, C > 0$. Is it possible to construct a triangle with sides of lengths $a, b, c$?

1941 Moscow Mathematical Olympiad, 082

Tags: combinatorial geometry , minimum , geometry , Triangle , Tiling

* Given $\vartriangle ABC$, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same $\vartriangle ABC$.

2007 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: combinatorial geometry , Triangle , Segment , combinatorics

Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments. Clarification : the vertices of the original triangle are not considered points of intersection or division.

2012 Brazil Team Selection Test, 4

Tags: trigonometry , triangle inequality , algebra , Triangle , IMO Shortlist

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]

2001 IMO Shortlist, 1

Tags: geometry , Triangle , IMO Shortlist , square

Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

2018 EGMO, 1

Tags: geometry , EGMO , Triangle , circle , EGMO 2018

Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$. Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.

1998 IMO Shortlist, 3

Tags: geometry , incenter , trigonometry , Triangle , IMO , IMO 1998

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.

2013 VTRMC, Problem 2

Tags: geometry , Triangle

Let $ABC$ be a right-angled triangle with $\angle ABC=90^\circ$, and let $D$ be on $AB$ such that $AD=2DB$. What is the maximum possible value of $\angle ACD$?

1959 Czech and Slovak Olympiad III A, 1

Tags: construction , Triangle , Medians

Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.

2019 Czech and Slovak Olympiad III A, 4

Tags: geometry , excircle , Triangle , perpendicular lines , national olympiad

Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.