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: 25757

2000 Belarusian National Olympiad, 8
Tags: geometry
To any triangle with side lengths $a,b,c$ and the corresponding angles $\alpha, \beta, \gamma$ (measured in radians), the 6-tuple $(a,b,c,\alpha, \beta, \gamma)$ is assigned. Find the minimum possible number $n$ of distinct terms in the 6-tuple assigned to a scalene triangle.

1997 Tuymaada Olympiad, 8
Tags: geometry , right triangle , congruent triangles
Find a right triangle that can be cut into $365$ equal triangles.

1990 China Team Selection Test, 2
Tags: geometry
Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin $O$ that cuts them both, then these polygons are called "properly placed". Find the least $m \in \mathbb{N}$, such that for any group of properly placed polygons, $m$ lines can drawn through $O$ and every polygon is cut by at least one of these $m$ lines.

1987 China Team Selection Test, 1
Tags: geometry , rectangle , ratio , analytic geometry , symmetry , hyperbola , conic
Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.

III Soros Olympiad 1996 - 97 (Russia), 9.3
Tags: geometry
Draw the set of projections of a square given on a plane onto all possible lines passing through a given point $O$ of the plane lying outside the square.

2011 China Team Selection Test, 1
Tags: geometry , circumcircle , trigonometry , trapezoid , geometric transformation , reflection , parallelogram
Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.

2002 Turkey MO (2nd round), 2
Tags: incenter , angle bisector , geometry , circumcircle
Two circles are externally tangent to each other at a point $A$ and internally tangent to a third circle $\Gamma$ at points $B$ and $C.$ Let $D$ be the midpoint of the secant of $\Gamma$ which is tangent to the smaller circles at $A.$ Show that $A$ is the incenter of the triangle $BCD$ if the centers of the circles are not collinear.

2011 Mexico National Olympiad, 5
Tags: geometry , rectangle , induction , combinatorics
A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.

May Olympiad L1 - geometry, 2000.2
Tags: geometry , right triangle , angle bisector , midpoint
Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side $AB$ .

2005 Sharygin Geometry Olympiad, 7
Tags: equilateral , equilateral triangle , concentric circles , angle , geometry
Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?

2013 Hanoi Open Mathematics Competitions, 7
Tags: geometry , area of a triangle , equilateral , area
Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.

2006 AIME Problems, 1
Tags: geometry , perimeter , trigonometry , pythagorean theorem
In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$.

1994 Taiwan National Olympiad, 1
Tags: cyclic quadrilateral , geometry
Let $ABCD$ be a quadrilateral with $AD=BC$ and $\widehat{A}+\widehat{B}=120^{0}$. Let us draw equilateral $ACP,DCQ,DBR$ away from $AB$ . Prove that the points $P,Q,R$ are collinear.

2018 Iran MO (1st Round), 25
Tags: sphere , geometry , physics
Astrophysicists have discovered a minor planet of radius $30$ kilometers whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water. This incidence causes an increase of $1$ centimeters to the height of the water on this planet. What is the radius of the meteor in meters?

2023 Novosibirsk Oral Olympiad in Geometry, 6
Tags: geometry , angle
Two quarter-circles touch as shown. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png[/img]

2022 Novosibirsk Oral Olympiad in Geometry, 6
Tags: equilateral , geometry
Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.

1997 Brazil Team Selection Test, Problem 1
Tags: triangle , geometry , circumcircle
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.

2004 Baltic Way, 19
Tags: circumcircle , geometric transformation , reflection , ratio , angle bisector , geometry
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$. Let $M$ be a point on the side $BC$ such that $\angle BAM = \angle DAC$. Further, let $L$ be the second intersection point of the circumcircle of the triangle $CAM$ with the side $AB$, and let $K$ be the second intersection point of the circumcircle of the triangle $BAM$ with the side $AC$. Prove that $KL \parallel BC$.

1967 IMO Shortlist, 5
Tags: geometry , 3d geometry , sphere , polyhedron
Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

2024 All-Russian Olympiad Regional Round, 9.1
Tags: geometry , rectangle
There are $2024$ rectangles $1 \times n$ for $n=1, 2, \ldots, 2024$. Is it possible to make a square using some of them, such that the side length of the square is greater than $1$?

2024 Sharygin Geometry Olympiad, 8.7
Tags: geometry
A convex quadrilateral $ABCD$ is given. A line $l \parallel AC$ meets the lines $AD$, $BC$, $AB$, $CD$ at points $X$, $Y$, $Z$, $T$ respectively. The circumcircles of triangles $XYB$ and $ZTB$ meet for the second time at point $R$. Prove that $R$ lies on $BD$.

2011 Saudi Arabia BMO TST, 3
Tags: bisects segment , equal angles , pentagon , geometry
Let $ABCDE$ be a convex pentagon such that $\angle BAC = \angle CAD = \angle DAE$ and $\angle ABC = \angle ACD = \angle ADE$. Diagonals $BD$ and $CE$ meet at $P$. Prove that $AP$ bisects side $CD$.

1998 All-Russian Olympiad Regional Round, 11.6
Tags: geometry , perimeter
A polygon with sides running along the sides of the squares was cut out of an endless chessboard. A segment of the perimeter of a polygon is called black if the polygon adjacent to it from the inside is which cell is black, respectively white if the cell is white. Let $A$ be the number of black segments on the perimeter, and $B$ be the number of white ones, Let the polygon consist of $a$ black and $b$ white cells. Prove that $A-B = 4(a -b)$.

2023 Stars of Mathematics, 1
Tags: combinatorics , geometry
A convex polygon is dissected into a finite number of triangles with disjoint interiors, whose sides have odd integer lengths. The triangles may have multiple vertices on the boundary of the polygon and their sides may overlap partially. [list=a] [*]Prove that the polygon's perimeter is an integer which has the same parity as the number of triangles in the dissection. [*]Determine whether part a) holds if the polygon is not convex. [/list] [i]Proposed by Marius Cavachi[/i] [i]Note: the junior version only included part a), with an arbitrary triangle instead of a polygon.[/i]

2013 Dutch BxMO/EGMO TST, 1
Tags: parallel , equal area , area , geometry
In quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.