Olympiad problems

1975 All Soviet Union Mathematical Olympiad, 211

Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons.

2022 239 Open Mathematical Olympiad, 3

Tags: color , geometry , combinatorial geometry , point

Let $A$ be a countable set, some of its countable subsets are selected such that; the intersection of any two selected subsets has at most one element. Find the smallest $k$ for which one can ensure that we can color elements of $A$ with $k$ colors such that each selected subsets exactly contain one element of one of the colors and an infinite number of elements of each of the other colors.

ICMC 5, 3

Tags: geometry , combinatorial geometry , 3d geometry , symmetry

A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry. [i]Proposed by Ethan Tan[/i]

III Soros Olympiad 1996 - 97 (Russia), 11.9

Tags: combinatorics , combinatorial geometry , geometry

Given a regular hexagon with a side of $100$. Each side is divided into one hundred equal parts. Through the division points and vertices of the hexagon, all sorts of straight lines parallel to its sides are drawn. These lines divided the hexagon into single regular triangles. Consider covering a hexagon with equal rhombuses. Each rhombus is made up of two triangles. (These rhombuses cover the entire hexagon and do not overlap.) Among the lines that form our grid, we select those that intersect exactly to the rhombuses (intersect diagonally). How many such lines will there be if: a) $k = 101$; b) $k = 100$; c) $k = 87$?

2003 Switzerland Team Selection Test, 5

Tags: game , combinatorial geometry , combinatorics , board

There are $n$ pieces on the squares of a $5 \times 9$ board, at most one on each square at any time during the game. A move in the game consists of simultaneously moving each piece to a neighboring square by side, under the restriction that a piece having been moved horizontally in the previous move must be moved vertically and vice versa. Find the greatest value of $n$ for which there exists an initial position starting at which the game can be continued until the end of the world.

1948 Moscow Mathematical Olympiad, 142

Tags: distance , combinatorics , combinatorial geometry

Find all possible arrangements of $4$ points on a plane, so that the distance between each pair of points is equal to either $a$ or $b$. For what ratios of $a : b$ are such arrangements possible?

May Olympiad L2 - geometry, 2019.5

Tags: combinatorial geometry , combinatorics , regular polygon

We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set? [hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]

1970 IMO Shortlist, 12

Tags: acute , triangle , point set , counting , combinatorial geometry

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

2013 Estonia Team Selection Test, 2

Tags: combinatorics , combinatorial geometry

For which positive integers $n \ge 3$ is it possible to mark $n$ points of a plane in such a way that, starting from one marked point and moving on each step to the marked point which is the second closest to the current point, one can walk through all the marked points and return to the initial one? For each point, the second closest marked point must be uniquely determined.

2004 All-Russian Olympiad, 1

Tags: geometry , geometric transformation , rotation , analytic geometry , combinatorics , combinatorial geometry

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

2016 Regional Olympiad of Mexico West, 6

Tags: coloring , combinatorics , combinatorial geometry

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.

1996 North Macedonia National Olympiad, 4

Tags: geometry , combinatorial geometry , polygon , parallel

A polygon is called [i]good [/i] if it satisfies the following conditions: (i) All its angles are in $(0,\pi)$ or in $(\pi ,2\pi)$, (ii) It is not self-intersecing, (iii) For any three sides, two are parallel and equal. Find all $n$ for which there exists a [i]good [/i] $n$-gon.

2010 Federal Competition For Advanced Students, P2, 6

Tags: geometry , circumcircle , hexagon , combinatorial geometry

A diagonal of a convex hexagon is called [i]long[/i] if it decomposes the hexagon into two quadrangles. Each pair of [i]long[/i] diagonals decomposes the hexagon into two triangles and two quadrangles. Given is a hexagon with the property, that for each decomposition by two [i]long[/i] diagonals the resulting triangles are both isosceles with the side of the hexagon as base. Show that the hexagon has a circumcircle.

1990 Tournament Of Towns, (266) 4

Tags: combinatorics , combinatorial geometry , tiling

A square board with dimensions $100 \times 100$ is divided into $10 000 $unit squares. One of the squares is cut out. Is it possible to cover the rest of the board by isosceles right angled triangles which have hypotenuses of length $2$, and in such a way that their hypotenuses lie on sides of the squares and their other two sides lie on diagonals? The triangles must not overlap each other or extend beyond the edges of the board. (S Fomin, Leningrad)

2017 IMAR Test, 4

Tags: combinatorial geometry , combinatorics , geometry , polygon

Let $n$ be an integer greater than or equal to $3$, and let $P_n$ be the collection of all planar (simple) $n$-gons no two distinct sides of which are parallel or lie along some line. For each member $P$ of $P_n$, let $f_n(P)$ be the least cardinal a cover of $P$ by triangles formed by lines of support of sides of $P$ may have. Determine the largest value $f_n(P)$ may achieve, as $P$ runs through $P_n$.

1996 Tournament Of Towns, (500) 2

Tags: combinatorics , combinatorial geometry

The square $0\le x\le 1$, $0\le y\le 1$ is drawn in the plane $Oxy$. A grasshopper sitting at a point $M$ with noninteger coordinates outside this square jumps to a new point which is symmetrical to $M$ with respect to the leftmost (from the grasshopper’s point of view) vertex of the square. Prove that no matter how many times the grasshopper jumps, it will never reach the distance more than $10 d$ from the center $C$ of the square, where $d$ is the distance between the initial position $M$ and the center $C$. (A Kanel)

2007 China Girls Math Olympiad, 4

Tags: symmetry , combinatorial geometry , 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.

KoMaL A Problems 2020/2021, A. 793

Tags: combinatorics , geometry , combinatorial geometry

In the $43$ dimension Euclidean space the convex hull of finite set $S$ contains polyhedron $P$. We know that $P$ has $47$ vertices. Prove that it is possible to choose at most $2021$ points in $S$ such that the convex hull of these points also contain $P$, and this is sharp.

2021 Caucasus Mathematical Olympiad, 3

Tags: combinatorial geometry , combinatorics

Let $n\ge 3$ be a positive integer. In the plane $n$ points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked. (Recall that the vertices of a triangle are not collinear.)

2018 Romania Team Selection Tests, 3

Tags: combinatorial geometry , combinatorics

Consider a 4-point configuration in the plane such that every 3 points can be covered by a strip of a unit width. Prove that: 1) the four points can be covered by a strip of length at most $\sqrt2$ and 2)if no strip of length less that $\sqrt2$ covers all the four points, then the points are vertices of a square of length $\sqrt2$

2001 China Team Selection Test, 1

Tags: graph theory , combinatorics , combinatorial geometry

Given seven points on a plane, with no three points collinear. Prove that it is always possible to divide these points into the vertices of a triangle and a convex quadrilateral, with no shared parts between the two shapes.

1990 Tournament Of Towns, (247) 1

Tags: combinatorial geometry , combinatorics

Find the maximum number of parts into which the $Oxy$-plane can be divided by $100$ graphs of different quadratic functions of the form $y = ax^2 + bx + c$. (N.B. Vasiliev, Moscow)

1990 Tournament Of Towns, (279) 4

Tags: geometry , combinatorics , combinatorial geometry

There are $20$ points in the plane and no three of them are collinear. Of these points $10$ are red while the other $10$ are blue. Prove that there exists a straight line such that there are $5$ red points and $5$ blue points on either side of this line. (A Kushnirenko, Moscow)

1998 All-Russian Olympiad Regional Round, 9.7

Tags: combinatorics , combinatorial geometry , geometry , regular polygon

Given a billiard in the form of a regular $1998$-gon $A_1A_2...A_{1998}$. A ball was released from the midpoint of side $A_1A_2$, which, reflected therefore from sides $A_2A_3$, $A_3A_4$, . . . , $A_{1998}A_1$ (according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular $1998$-gon.

2004 Polish MO Finals, 5

Tags: geometry , combinatorial geometry

Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.