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: 77

Kyiv City MO Seniors 2003+ geometry, 2019.10.3
Tags: geometry , Integers , Triangles
Call a right triangle $ABC$ [i]special [/i] if the lengths of its sides $AB, BC$ and$ CA$ are integers, and on each of these sides has some point $X$ (different from the vertices of $ \vartriangle ABC$), for which the lengths of the segments $AX, BX$ and $CX$ are integers numbers. Find at least one special triangle. (Maria Rozhkova)

1977 Yugoslav Team Selection Test, Problem 3
Tags: geometry , Triangles
Assume that the equality $2BC=AB+AC$ holds in $\triangle ABC$. Prove that: (a) The vertex $A$, the midpoints $M$ and $N$ of $AB$ and $AC$ respectively, the incenter $I$, and the circumcenter $O$ belong to a circle $k$. (b) The line $GI$, where $G$ is the centroid of $\triangle ABC$ is a tangent to $k$.

2001 Moldova National Olympiad, Problem 4
Tags: geometry , Triangles
In a triangle $ABC$, $BC=a$, $AC=b$, $\angle B=\beta$ and $\angle C=\gamma$. Prove that the bisector of the angle at $A$ is equal to the altitude from $B$ if and only if $b=a\cos\frac{\beta-\gamma}2$.

1979 Bulgaria National Olympiad, Problem 3
Tags: combinatorics , geometry , combinatorial geometry , Triangles
Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.

1997 Brazil Team Selection Test, Problem 5
Tags: geometry , geometric inequality , inequalities , Geometric Inequalities , Triangles , Triangle , area
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

1982 Bulgaria National Olympiad, Problem 6
Tags: Triangles , geometry , square , Centroid , triangle centers , Locus
Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.

1974 Bulgaria National Olympiad, Problem 5
Tags: Triangles , geometry
Find all point $M$ lying into given acute-angled triangle $ABC$ and such that the area of the triangle with vertices on the feet of the perpendiculars drawn from $M$ to the lines $BC$, $CA$, $AB$ is maximal. [i]H. Lesov[/i]

1983 Polish MO Finals, 1
Tags: combinatorial geometry , combinatorics , polygon , Triangles
On the plane are given a convex $n$-gon $P_1P_2....P_n$ and a point $Q$ inside it, not lying on any of its diagonals. Prove that if $n$ is even, then the number of triangles $P_iP_jP_k$ containing the point $Q$ is even.

1966 Czech and Slovak Olympiad III A, 4
Tags: geometry , 3D geometry , tetrahedron , Triangles
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.

1985 Bulgaria National Olympiad, Problem 5
Tags: Triangles , geometry
Let $P$ be a point on the median $CM$ of a triangle $ABC$ with $AC\ne BC$ and the acute angle $\gamma$ at $C$, such that the bisectors of $\angle PAC$ and $\angle PBC$ intersect at a point $Q$ on the median $CM$. Determine $\angle APB$ and $\angle AQB$.

1996 Estonia National Olympiad, 5
Tags: combinatorial geometry , combinatorics , Triangles
Suppose that $n$ triangles are given in the plane such that any three of them have a common vertex, but no four of them do. Find the greatest possible $n$.

1987 Bulgaria National Olympiad, Problem 5
Tags: geometry , Triangles
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 $\angle BAC$.

1997 Brazil Team Selection Test, Problem 1
Tags: geometry , circumcircle , Triangles
Let $ABC$ be a triangle and $L$ its circumscribed circle. The internal bisector of angle $A$ meets $BC$ at point $P$. Let $L_1$ be the circle tangent to $AP,BP$ and $L$. Similarly, let $L_2$ be the circle tangent to $AP,CP$ and $L$. Prove that the tangency points of $L_1$ and $L_2$ with $AP$ coincide.

2019 AMC 10, 16
Tags: AMC , AMC 10 , 2019 AMC 10B , Triangles , geometry
In $\triangle ABC$ with a right angle at $C,$ point $D$ lies in the interior of $\overline{AB}$ and point $E$ lies in the interior of $\overline{BC}$ so that $AC=CD,$ $DE=EB,$ and the ratio $AC:DE=4:3.$ What is the ratio $AD:DB?$ $\textbf{(A) } 2:3 \qquad\textbf{(B) } 2:\sqrt{5} \qquad\textbf{(C) } 1:1 \qquad\textbf{(D) } 3:\sqrt{5} \qquad\textbf{(E) } 3:2$

2003 Federal Math Competition of S&M, Problem 3
Tags: geometry , Triangles
Let $a,b$ and $c$ be the lengths of the edges of a triangle whose angles are $\alpha=40^\circ,\beta=60^\circ$ and $\gamma=80^\circ$. Prove that $$a(a+b+c)=b(b+c).$$

1981 Bulgaria National Olympiad, Problem 2
Tags: geometry , Triangles , angle
Let $ABC$ be a triangle such that the altitude $CH$ and the sides $CA,CB$ are respectively equal to a side and two distinct diagonals of a regular heptagon. Prove that $\angle ACB<120^\circ$.

2020 Jozsef Wildt International Math Competition, W9
Tags: inequalities , geometry , inradius , Triangle , Triangles
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]

1997 Brazil Team Selection Test, Problem 5
Tags: geometry , geometric inequality , inequalities , Geometric Inequalities , Triangles , Triangle , area
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

1966 Bulgaria National Olympiad, Problem 3
Tags: geometry , Triangles
(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 $B$.

1974 Yugoslav Team Selection Test, Problem 2
Tags: geometry , transformations , Triangles
Given two directly congruent triangles $ABC$ and $A'B'C'$ in a plane, assume that the circles with centers $C$ and $C'$ and radii $CA$ and $C'A'$ intersect. Denote by $\mathcal M$ the transformation that maps $\triangle ABC$ to $\triangle A'B'C'$. Prove that $\mathcal M$ can be expressed as a composition of at most three rotations in the following way: The first rotation has the center in one of $A,B,C$ and maps $\triangle ABC$ to $\triangle A_1B_1C_1$; The second rotation has the center in one of $A_1,B_1,C_1$, and maps $\triangle A_1B_1C_1$ to $\triangle A_2B_2C_2$; The third rotation has the center in one of $A_2,B_2,C_2$ and maps $\triangle A_2B_2C_2$ to $\triangle A'B'C'$.

1997 Brazil Team Selection Test, Problem 1
Tags: geometry , circumcircle , Triangles
Let $ABC$ be a triangle and $L$ its circumscribed circle. The internal bisector of angle $A$ meets $BC$ at point $P$. Let $L_1$ be the circle tangent to $AP,BP$ and $L$. Similarly, let $L_2$ be the circle tangent to $AP,CP$ and $L$. Prove that the tangency points of $L_1$ and $L_2$ with $AP$ coincide.

2021 ELMO Problems, 1
Tags: geometry , Elmo , Triangles , circles , Cyclic
In $\triangle ABC$, points $P$ and $Q$ lie on sides $AB$ and $AC$, respectively, such that the circumcircle of $\triangle APQ$ is tangent to $BC$ at $D$. Let $E$ lie on side $BC$ such that $BD = EC$. Line $DP$ intersects the circumcircle of $\triangle CDQ$ again at $X$, and line $DQ$ intersects the circumcircle of $\triangle BDP$ again at $Y$. Prove that $D$, $E$, $X$, and $Y$ are concyclic.

2014 Canadian Mathematical Olympiad Qualification, 6
Tags: geometry , Triangles , midpoints
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 $PQ$.

1992 Yugoslav Team Selection Test, Problem 1
Tags: geometry , Triangles
Three squares $BCDE,CAFG$ and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is an isosceles right triangle.

2001 Moldova National Olympiad, Problem 7
Tags: geometry , Triangles
The incircle of a triangle $ABC$ is centered at $I$ and touches $AC,AB$ and $BC$ at points $K,L,M$, respectively. The median $BB_1$ of $\triangle ABC$ intersects $MN$ at $D$. Prove that the points $I,D,K$ are collinear.