Olympiad problems

1995 Romania Team Selection Test, 2

Tags: combinatorics , combinatorial geometry , parallelepiped , volume , cube , partition

A cube is partitioned into finitely many rectangular parallelepipeds with the edges parallel to the edges of the cube. Prove that if the sum of the volumes of the circumspheres of these parallelepipeds equals the volume of the circumscribed sphere of the cube, then all the parallelepipeds are cubes.

1994 Tuymaada Olympiad, 8

Tags: geometry , analytic geometry , 3d geometry , sphere , lattice points , combinatorial geometry

Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.

2010 Federal Competition For Advanced Students, P2, 5

Tags: geometry , combinatorics , combinatorial geometry , rectangle

Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other. How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?

2023 Durer Math Competition Finals, 14

Tags: geometry , combinatorial geometry , combinatorics

Zeus’s lightning is made of a copper rod of length $60$ by bending it $4$ times in alternating directions so that the angle between two adjacent parts is always $60^o$. What is the minimum value of the square of the distance between the two endpoints of the lightning? All five segments of the lightning lie in the same plane. [img]https://cdn.artofproblemsolving.com/attachments/5/1/a18206df4fde561421022c0f2b4332f5ac44a2.png[/img]

2016 Regional Olympiad of Mexico West, 6

Tags: combinatorial geometry , coloring , combinatorics

The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.

2009 Chile National Olympiad, 6

Tags: geometry , combinatorics , combinatorial geometry , point

There are $n \ge 6$ green points in the plane, such that no $3$ of them are collinear. Suppose further that $6$ of these points are the vertices of a convex hexagon. Prove that there are $5$ green points that form a pentagon that does not contain any other green point inside.

Russian TST 2018, P1

Tags: geometry , combinatorics , combinatorial geometry

Find all positive $r{}$ satisfying the following condition: For any $d > 0$, there exist two circles of radius $r{}$ in the plane that do not contain lattice points strictly inside them and such that the distance between their centers is $d{}$.

2017 Bulgaria EGMO TST, 1

Tags: combinatorics , polygon , combinatorial geometry , triangulation

Prove that every convex polygon has at most one triangulation consisting entirely of acute triangles.

1985 All Soviet Union Mathematical Olympiad, 411

Tags: cube , combinatorial geometry , geometry , combinatorics , coloring , parallelepiped

The parallelepiped is constructed of the equal cubes. Three parallelepiped faces, having the common vertex are painted. Exactly half of all the cubes have at least one face painted. What is the total number of the cubes?

2011 Kyiv Mathematical Festival, 5

Tags: geometry , combinatorial geometry

Pete claims that he can draw $3$ segments of length $1$ and a circle of radius less than $\sqrt3 / 3$ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $3 $ segments. Is Pete right?

1982 IMO Longlists, 10

Tags: sphere , homothety , geometry , combinatorial geometry , 3d geometry , geometric transformation

Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that \[\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.\]

1997 Tournament Of Towns, (525) 2

Tags: combinatorics , combinatorial geometry , geometry

Baron Munchausen plays billiards on a table with the shape of an equilateral triangle. He claims to have shot a ball from one of the sides of this table so that it passed through a certain point three times in three different directions and then returned to the original point on the side. Can that be true, assuming that the usual law of reflection holds? (Μ Evdokimov)

2022 Novosibirsk Oral Olympiad in Geometry, 5

Tags: combinatorial geometry , geometry

Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.

2021 China Team Selection Test, 6

Tags: combinatorial geometry , combinatorics , geometry

Proof that there exist constant $\lambda$, so that for any positive integer $m(\ge 2)$, and any lattice triangle $T$ in the Cartesian coordinate plane, if $T$ contains exactly one $m$-lattice point in its interior(not containing boundary), then $T$ has area $\le \lambda m^3$. PS. lattice triangles are triangles whose vertex are lattice points; $m$-lattice points are lattice points whose both coordinates are divisible by $m$.

2017 Peru IMO TST, 2

Tags: combinatorial geometry , combinatorics

Let $n\geq3$ an integer. Mario draws $20$ lines in the plane, such that there are not two parallel lines. For each [b]equilateral triangle[/b] formed by three of these lines, Mario receives three coins. For each [b]isosceles[/b] and [b]non-equilateral[/b] triangle ([u]at the same time[/u]) formed by three of these lines, Mario receives a coin. How is the maximum number of coins that can Mario receive?

2015 Romania Masters in Mathematics, 6

Tags: combinatorial geometry , combinatorics , rectangle

Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every set $C$ of $4n$ points in the interior of the unit square $U$, there exists a rectangle $T$ contained in $U$ such that $\bullet$ the sides of $T$ are parallel to the sides of $U$; $\bullet$ the interior of $T$ contains exactly one point of $C$; $\bullet$ the area of $T$ is at least $\mu$.

1980 Kurschak Competition, 1

Tags: combinatorial geometry , combinatorics , coloring

The points of space are coloured with five colours, with all colours being used. Prove that some plane contains four points of different colours.

2009 Germany Team Selection Test, 1

Tags: combinatorial geometry , rectangle , extremal combinatorics , combinatorics , geometry

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

2016 Singapore Senior Math Olympiad, 2

Tags: combinatorial geometry , point , line , min , combinatorics

Let $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.

1948 Moscow Mathematical Olympiad, 155

Tags: combinatorics , geometry , angle , combinatorial geometry , obtuse , ray , maximum

What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?

Kvant 2020, M2602

Tags: geometry , combinatorial geometry

For a given natural number $k{}$, a convex polygon is called $k{}$[i]-triangular[/i] if it is the intersection of some $k{}$ triangles. [list=a] [*]What is the largest $n{}$ for which there exist a $k{}$-triangular $n{}$-gon? [*]What is the largest $n{}$ for which any convex $n{}$-gon is $k{}$-triangular? [/list] [i]Proposed by P. Kozhevnikov[/i]

2024 Brazil National Olympiad, 3

Tags: geometry , angle bisector , combinatorial geometry , diagonal

Let $ n \geq 3 $ be a positive integer. In a convex polygon with $ n $ sides, all the internal bisectors of its $ n $ internal angles are drawn. Determine, as a function of $ n $, the smallest possible number of distinct lines determined by these bisectors.

2012 Belarus Team Selection Test, 1

Tags: geometry , combinatorial geometry , regular polygon , regular , combinatorics

Let $m,n,k$ be pairwise relatively prime positive integers greater than $3$. Find the minimal possible number of points on the plane with the following property: there are $x$ of them which are the vertices of a regular $x$-gon for $x = m, x = n, x = k$. (E.Piryutko)

1986 Tournament Of Towns, (131) 7

Tags: circles , combinatorial geometry , combinatorics , point , arc , angle

On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

2007 Cuba MO, 2

Tags: combinatorics , combinatorial geometry , geometry , prism , 3d geometry

A prism is called [i]binary [/i] if it can be assigned to each of its vertices a number from the set $\{-1, 1\}$, such that the product of the numbers assigned to the vertices of each face is equal to $-1$. a) Prove that the number of vertices of the binary prisms is divisible for $8$. b) Prove that a prism with $2000$ vertices is binary.