Olympiad problems

1972 All Soviet Union Mathematical Olympiad, 160

Tags: combinatorial geometry

Given $50$ segments on the line. Prove that one of the following statements is valid: 1. Some $8$ segments have the common point. 2. Some $8$ segments do not intersect each other.

1934 Eotvos Mathematical Competition, 3

Tags: combinatorial geometry , combinatorics

We are given an infinite set of rectangles in the plane, each with vertices of the form $(0, 0)$, $(0,m)$, $(n, 0)$ and $ (n,m)$, where $m$ and $n$ are positive integers. Prove that there exist two rectangles in the set such that one contains the other.

2007 China Girls Math Olympiad, 4

Tags: combinatorial geometry , symmetry , combinatorics

The set $ S$ consists of $ n > 2$ points in the plane. The set $ P$ consists of $ m$ lines in the plane such that every line in $ P$ is an axis of symmetry for $ S$. Prove that $ m\leq n$, and determine when equality holds.

1978 All Soviet Union Mathematical Olympiad, 259

Tags: combinatorics , combinatorial geometry , square

Prove that there exists such a number $A$ that you can inscribe $1978$ different size squares in the plot of the function $y = A sin(x)$. (The square is inscribed if all its vertices belong to the plot.)

2019 Estonia Team Selection Test, 11

Tags: geometry , combinatorial geometry , circumcircle , perimeter

Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply: (a) the circumcircle of each triangle in the set $T$ is $\omega$; (b) The interior of any two triangles in the set $T$ has no common point. Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.

2020 Ukrainian Geometry Olympiad - December, 2

Tags: combinatorics , combinatorial geometry , acute , geometry

On a circle noted $n$ points. It turned out that among the triangles with vertices in these points exactly half of the acute. Find all values $n$ in which this is possible.

2008 Indonesia TST, 1

Tags: combinatorial geometry , combinatorics , area of a triangle , geometry

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

2015 Caucasus Mathematical Olympiad, 3

Tags: combinatorial geometry , combinatorics , tiles , tiling

The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each other.)

2006 All-Russian Olympiad Regional Round, 10.1

Tags: geometry , combinatorics , combinatorial geometry , geometric inequality

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.

1969 Yugoslav Team Selection Test, Problem 6

Tags: geometry , combinatorics , combinatorial geometry

Let $E$ be the set of $n^2+1$ closed intervals on the real axis. Prove that there exists a subset of $n+1$ intervals that are monotonically increasing with respect to inclusion, or a subset of $n+1$ intervals none of which contains any other interval from the subset.

2000 Chile National Olympiad, 6

Tags: combinatorics , tiling , tiles , rectangle , combinatorial geometry , geometry

With $76$ tiles, of which some are white, other blue and the remaining red, they form a rectangle of $4 \times 19$. Show that there is a rectangle, inside the largest, that has its vertices of the same color.

1988 Poland - Second Round, 5

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

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

1965 All Russian Mathematical Olympiad, 064

Tags: combinatorial geometry , combinatorics , unit square

Is it possible to place $1965$ points in a square with side $1$ so that any rectangle of area $1/200$ with sides parallel to the sides of the square contains at least one of these points inside?

1992 Tournament Of Towns, (357) 6

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

Consider a polyhedron having $100$ edges. (a) Find the maximal possible number of its edges which can be intersected by a plane (not containing any vertices of the polyhedron) if the polyhedron is convex. (b) Prove that for a non-convex polyhedron this number i. can be as great as $96$, ii. cannot be as great as $100$. (A Andjans, Riga

2024 Sharygin Geometry Olympiad, 8.8

Tags: geometry , combinatorial geometry

Two polygons are cut from the cardboard. Is it possible that for any disposition of these polygons on the plane they have either common inner points or only a finite number of common points on the boundary?

2008 China Team Selection Test, 3

Tags: polynomial , combinatorial geometry , geometry , algebra , gauss , complex number

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.

2002 Switzerland Team Selection Test, 4

Tags: combinatorial geometry , combinatorics

A $7 \times 7$ square is divided into unit squares by lines parallel to its sides. Some Swiss crosses (obtained by removing corner unit squares from a square of side $3$) are to be put on the large square, with the edges along division lines. Find the smallest number of unit squares that need to be marked in such a way that every cross covers at least one marked square.

1997 Tournament Of Towns, (565) 6

Tags: geometry , combinatorics , combinatorial geometry , congruent triangles

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into n equal segments and the triangle into n congruent triangles. Each of these n triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. (a) If $n =10$, what is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (b)The same question for $n = 9$. (R Zhenodarov)

2020/2021 Tournament of Towns, P7

Tags: combinatorial geometry , combinatorics

An integer $n > 2$ is given. Peter wants to draw $n{}$ arcs of length $\alpha{}$ of great circles on a unit sphere so that they do not intersect each other. Prove that [list=a] [*]for all $\alpha<\pi+2\pi/n$ it is possible; [*]for all $\alpha>\pi+2\pi/n$ it is impossible; [/list] [i]Ilya Bogdanov[/i]

VI Soros Olympiad 1999 - 2000 (Russia), 10.7

Tags: combinatorial geometry , combinatorics

The numbers $1, 2, 3, ..., 99, 100$ are randomly arranged in the cells of a square table measuring $10\times 10$ (each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 1$82$. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.

1982 Tournament Of Towns, (028) 2

Tags: geometry , combinatorial geometry , polyhedron , 3d geometry

Does there exist a polyhedron (not necessarily convex) which could have the following complete list of edges? $AB, AC, BC, BD, CD, DE, EF, EG, FG, FH, GH, AH$. [img]http://1.bp.blogspot.com/-wTdNfQHG5RU/XVk1Bf4wpqI/AAAAAAAAKhA/8kc6u9KqOgg_p1CXim2LZ1ANFXFiWgnYACK4BGAYYCw/s1600/TOT%2B1982%2BAutum%2BS2.png[/img]

1991 ITAMO, 5

Tags: combinatorial geometry , geometry , polyhedron , 3d geometry

For which values of $n$ does there exist a convex polyhedron with $n$ edges?

1997 Tournament Of Towns, (556) 6

Tags: geometry , combinatorial geometry , equilateral , combinatorics

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into $10$ equal segments and the triangle into $100$ congruent triangles. Each of these $100$ triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. What is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe? (R Zhenodarov)

1976 Poland - Second Round, 3

Tags: geometry , combinatorial geometry , 3d geometry , sphere

We consider a spherical bowl without any great circle. The distance between points $A$ and $B$ on such a bowl is defined as the length of the arc of the great circle of the sphere with ends at points $A$ and $B$, which is contained in the bowl. Prove that there is no isometry mapping this bowl to a subset of the plane. Attention. A spherical bowl is each of the two parts into which the surface of the sphere is divided by a plane intersecting the sphere.

KoMaL A Problems 2017/2018, A. 710

Tags: combinatorics , combinatorial geometry

For which $n{}$ can we partition a regular $n{}$-gon into finitely many triangles such that no two triangles share a side? [i]Based on a problem of the 2017 Miklós Schweitzer competition[/i]