Olympiad problems

1983 Tournament Of Towns, (036) O5

A version of billiards is played on a right triangular table, with a pocket in each of the three corners, and one of the acute angles being $30^o$. A ball is played from just in front of the pocket at the $30^o$. vertex toward the midpoint of the opposite side. Prove that if the ball is played hard enough, it will land in the pocket of the $60^o$ vertex after $8$ reflections.

2004 Austria Beginners' Competition, 4

Tags: geometry , rhombus , construction

Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus.

2014 Sharygin Geometry Olympiad, 14

Tags: geometry , area , circles

In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area.

2019 Estonia Team Selection Test, 6

Tags: lattice , combinatorics , geometry , rhombus

It is allowed to perform the following transformations in the plane with any integers $a$: (1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$, (2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$. Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to: a) Vertices of a square, b) Vertices of a rectangle with unequal side lengths?

2021 Sharygin Geometry Olympiad, 16

Tags: geometry , circles

Let circles $\Omega$ and $\omega$ touch internally at point $A$. A chord $BC$ of $\Omega$ touches $\omega$ at point $K$. Let $O$ be the center of $\omega$. Prove that the circle $BOC$ bisects segment $AK$.

2002 Tournament Of Towns, 2

Tags: geometric transformation , reflection , rotation , geometry

$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.

2023 Iranian Geometry Olympiad, 4

Tags: geometry , concurrency

Let $ABC$ be a triangle and $P$ be the midpoint of arc $BAC$ of circumcircle of triangle $ABC$ with orthocenter $H$. Let $Q, S$ be points such that $HAPQ$ and $SACQ$ are parallelograms. Let $T$ be the midpoint of $AQ$, and $R$ be the intersection point of the lines $SQ$ and $PB$. Prove that $AB$, $SH$ and $TR$ are concurrent. [i]Proposed by Dominik Burek - Poland[/i]

2019 PUMaC Team Round, 5

Tags: floor function , analytic geometry , geometry

Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.

2007 Pan African, 3

Tags: geometry

An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?

1989 IMO Longlists, 4

Tags: geometry , circumcircle , trigonometry , parallelogram , rhombus , angle

The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$

2021 Romania National Olympiad, 1

Tags: geometry , perpendicular bisector , circles

Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre. [i]Ion Pătrașcu, Ion Cotoi[/i]

2008 China Team Selection Test, 3

Tags: analytic geometry , modular arithmetic , trapezoid , induction , trigonometry , number theory , geometry

Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)

1980 IMO Longlists, 4

Tags: geometry , circumcircle , algebra , polynomial , convex polygon

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

2002 Croatia National Olympiad, Problem 3

Tags: geometry , triangle

If two triangles with side lengths $a,b,c$ and $a',b',c'$ and the corresponding angle $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$ satisfy $\alpha+\alpha'=\pi$ and $\beta=\beta'$, prove that $aa'=bb'+cc'$.

1952 Miklós Schweitzer, 1

Tags: 3d geometry , pyramid , tetrahedron , convex , geometry

Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).

2006 Estonia Math Open Junior Contests, 8

Tags: geometry

Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point P which lies outside each circle. The first line intersects the first circle at A and A′ and the second circle at B and B′; here A and B are closer to P than A′ and B′, respectively, and P lies on segment AB. Analogously, the second line intersects the first circle at C and C′ and the second circle at D and D′. Prove that the points A, B, C, D are concyclic if and only if the points A′, B′, C′, D′ are concyclic.

2020/2021 Tournament of Towns, P1

Tags: geometry

Is it possible to select 100 points on a circle so that there are exactly 1000 right triangles with the vertices at selected points? [i]Sergey Dvoryaninov[/i]

2005 Colombia Team Selection Test, 1

Tags: geometry , 3d geometry , number theory

Let $a,b,c$ be integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$ prove that $abc$ is a perfect cube!

1988 Poland - Second Round, 5

Tags: geometry , rectangle , geometric inequalities , combinatorial geometry , combinatorics

Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.

1992 Poland - Second Round, 5

Tags: geometry , 3d geometry , sphere , geometric inequalities , tetrahedron

Determine the upper limit of the volume of spheres contained in tetrahedra of all heights not longer than $ 1 $.

2019 District Olympiad, 3

Tags: 3d geometry , geometry , parallelepiped , angle

Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.

1957 AMC 12/AHSME, 7

Tags: geometry , perimeter , inradius

The area of a circle inscribed in an equilateral triangle is $ 48\pi$. The perimeter of this triangle is: $ \textbf{(A)}\ 72\sqrt{3} \qquad \textbf{(B)}\ 48\sqrt{3}\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 72$

1985 Greece National Olympiad, 1

Tags: geometry , vector , analytic geometry , area

Inside triangle $ABC$ consider random point $O$. Prove that: $$E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O}$$ where $E_A,E_B,E_C$ the areas of triangle $BOC, COB, AOB$ respectively

2022 Brazil National Olympiad, 3

Tags: geometry

Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.

1987 ITAMO, 2

Tags: 3d geometry , tetrahedron , geometry

A tetrahedron has the property that the three segments connecting the pairs of midpoints of opposite edges are equal and mutually orthogonal. Prove that this tetrahedron is regular.