This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 405

1990 Bulgaria National Olympiad, Problem 5
Tags: triangle , geometry , arc , circles
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.

2013 Singapore MO Open, 5
Tags: perimeter , triangle , geometry , calculus
Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.

1967 IMO Shortlist, 5
Tags: counting , triangle , triangle inequality , combinatorics
Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$

1966 Czech and Slovak Olympiad III A, 4
Tags: tetrahedron , 3d geometry , triangle , geometry
Two triangles $ABC,ABD$ (with the common side $c=AB$) are given in space. Triangle $ABC$ is right with hypotenuse $AB$, $ABD$ is equilateral. Denote $\varphi$ the dihedral angle between planes $ABC,ABD$. 1) Determine the length of $CD$ in terms of $a=BC,b=CA,c$ and $\varphi$. 2) Determine all values of $\varphi$ such that the tetrahedron $ABCD$ has four sides of the same length.

2005 IMO, 1
Tags: concurrency , triangle , geometry , ratio
Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths. Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent. [i]Bogdan Enescu, Romania[/i]

2016 Brazil Team Selection Test, 1
Tags: triangle , geometry
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 $IJ = AH$.

2021 Belarusian National Olympiad, 10.2
Tags: triangle , geometry
In a triangle $ABC$ equality $2BC=AB+AC$ holds. The angle bisector of $\angle BAC$ inteesects $BC$ at $L$. A circle, that is tangent to $AL$ at $L$ and passes through $B$ intersects $AB$ for the second time at $X$. A circle, that is tangent to $AL$ at $L$ and passes through $C$ intersects $AC$ for the second time at $Y$ Find all possible values of $XY:BC$

1972 IMO Longlists, 8
Tags: triangle , point set , geometry , combinatorial geometry , combinatorics
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 $A_i.$

2007 IMO Shortlist, 1
Tags: triangle , incenter , geometry , circumcircle
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]

1978 IMO Longlists, 41
Tags: circumcircle , triangle , geometry , inradius , incenter
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 $ABC.$

2021 ISI Entrance Examination, 6
Tags: triangle , geometry
If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.

2020 Germany Team Selection Test, 2
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)

2009 Sharygin Geometry Olympiad, 4
Tags: geometry , triangle , regular polygon
Given regular $17$-gon $A_1 ... A_{17}$. Prove that two triangles formed by lines $A_1A_4, A_2A_{10}, A_{13}A_{14}$ and $A_2A_3, A_4A_6 A_{14}A_{15} $ are equal. (N.Beluhov)

1999 Brazil Team Selection Test, Problem 3
Tags: triangle , geometry
Let $BD$ and $CE$ be the bisectors of the interior angles $\angle B$ and $\angle C$, respectively ($D\in AC$, $E\in AB$). Consider the circumcircle of $ABC$ with center $O$ and the excircle corresponding to the side $BC$ with center $I_a$. These two circles intersect at points $P$ and $Q$. (a) Prove that $PQ$ is parallel to $DE$. (b) Prove that $I_aO$ is perpendicular to $DE$.

1972 IMO Shortlist, 2
Tags: triangle , point set , combinatorial geometry , combinatorics , geometry
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 Bundeswettbewerb Mathematik, 3
Tags: similar , circumcircle , triangle , geometry
In a triangle $ABC$, points $P, Q$ and $ R$ distinct from the vertices of the triangle are chosen on sides $AB, BC$ and $CA$, respectively. The circumcircles of the triangles $APR$, $BPQ$, and $CQR$ are drawn. Prove that the centers of these circles are the vertices of a triangle similar to triangle $ABC$.

1990 French Mathematical Olympiad, Problem 5
Tags: triangle , geometry
In a triangle $ABC$, $\Gamma$ denotes the excircle corresponding to $A$, $A',B',C'$ are the points of tangency of $\Gamma$ with $BC,CA,AB$ respectively, and $S(ABC)$ denotes the region of the plane determined by segments $AB',AC'$ and the arc $C'A'B'$ of $\Gamma$. Prove that there is a triangle $ABC$ of a given perimeter $p$ for which the area of $S(ABC)$ is maximal. For this triangle, give an approximate measure of the angle at $A$.

2014 Canadian Mathematical Olympiad Qualification, 6
Tags: triangle , midpoint , geometry
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 $PQ$.

1964 Czech and Slovak Olympiad III A, 4
Tags: geometric construction , equilateral , triangle , geometry
Points $A, S$ are given in plane such that $AS = a > 0$ as well as positive numbers $b, c$ satisfying $b < a < c$. Construct an equilateral triangle $ABC$ with the property $BS = b$, $CS = c$. Discuss conditions of solvability.

2018 Pan-African Shortlist, G5
Tags: triangle , parallel lines , incircle , geometry
Let $ABC$ be a triangle with $AB \neq AC$. The incircle of $ABC$ touches the sides $BC$, $CA$, $AB$ at $X$, $Y$, $Z$ respectively. The line through $Z$ and $Y$ intersects $BC$ extended in $X^\prime$. The lines through $B$ that are parallel to $AX$ and $AC$ intersect $AX^\prime$ in $K$ and $L$ respectively. Prove that $AK = KL$.

2019 IMO Shortlist, G4
Tags: triangle , geometry
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 France Team Selection Test, 3
Tags: midpoint , triangle , congruent triangles , geometry , incenter
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.

1994 Canada National Olympiad, 5
Tags: angle , triangle , geometry , trigonometry
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.

2007 German National Olympiad, 4
Tags: angle , triangle , arithmetic sequence , geometry
Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.

2019 India PRMO, 6
Tags: triangle , geometry
Let $ABC$ be a triangle such that $AB=AC$. Suppose the tangent to the circumcircle of ABC at B is perpendicular to AC. Find angle ABC measured in degrees