Olympiad problems

2009 Argentina National Olympiad, 1

$2009$ points have been marked on a circle. Lucía colors them with $7$ different colors of her choice. Then Ivan can join three points of the same color, thus forming monochrome triangles. Triangles cannot have points in common; not even vertices in common. Ivan's goal is to draw as many monochrome triangles as possible. Lucía's objective is to prevent Iván's task as much as possible through a good choice of colouring. How many monochrome triangles will Ivan get if they both do their homework to the best of their ability?

1999 Cono Sur Olympiad, 6

Tags: geometry , rotation , combinatorial geometry

An ant walks across the floor of a circular path of radius $r$ and moves in a straight line, but sometimes stops. Each time it stops, before resuming the march, it rotates $60^o$ alternating the direction (if the last time it turned $60^o$ to its right, the next one does it $60^o$ to its left, and vice versa). Find the maximum possible length of the path the ant goes through. Prove that the length found is, in fact, as long as possible. Figure: turn $60^o$ to the right .

2014 Online Math Open Problems, 25

Tags: induction , combinatorial geometry , factorial

Kevin has a set $S$ of $2014$ points scattered on an infinitely large planar gameboard. Because he is bored, he asks Ashley to evaluate \[ x = 4f_4 + 6f_6 + 8f_8 + 10f_{10} + \cdots \] while he evaluates \[ y = 3f_3 + 5f_5+7f_7+9f_9 + \cdots, \] where $f_k$ denotes the number of convex $k$-gons whose vertices lie in $S$ but none of whose interior points lie in $S$. However, since Kevin wishes to one-up everything that Ashley does, he secretly positions the points so that $y-x$ is as large as possible, but in order to avoid suspicion, he makes sure no three points lie on a single line. Find $\left\lvert y-x \right\rvert$. [i]Proposed by Robin Park[/i]

2006 Tournament of Towns, 5

Tags: polygon , max , non-convex , combinatorics , combinatorial geometry

A square is dissected into $n$ congruent non-convex polygons whose sides are parallel to the sides of the square, and no two of these polygons are parallel translates of each other. What is the maximum value of $n$? (4)

2019 Baltic Way, 10

Tags: combinatorics , combinatorial geometry

There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$, such that for any initial configuration of points it is possible to draw $k$ discs with the above property?

1952 Moscow Mathematical Olympiad, 227

Tags: plane , lines in plane , combinatorial geometry

$99$ straight lines divide a plane into $n$ parts. Find all possible values of $n$ less than $199$.

2018 Costa Rica - Final Round, 1

Tags: combinatorial geometry , geometry , combinatorics

There are $10$ points on a circle and all possible segments are drawn on the which two of these points are the endpoints. Determine the probability that selecting two segments randomly, they intersect at some point (it could be on the circumference).

2013 Abels Math Contest (Norwegian MO) Final, 2

Tags: geometry , combinatorial geometry , combinatorics , distance

In a triangle $T$, all the angles are less than $90^o$, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the distance from $p$ to $h$ is less than or equal to $s/\sqrt3$.

Novosibirsk Oral Geo Oly VII, 2020.7

Tags: congruent triangles , combinatorial geometry , geometry

The segments connecting the interior point of a convex non-sided $n$-gon with its vertices divide the $n$-gon into $n$ congruent triangles. For what is the smallest $n$ that is possible?

2024 239 Open Mathematical Olympiad, 2

Tags: combinatorial geometry , combinatorics

There are $2n$ points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose $n$ drawn segments having no common ends?

IV Soros Olympiad 1997 - 98 (Russia), 9.6

Tags: combinatorial geometry , geometry , combinatorics

Cut an acute triangle, one of whose sides is equal to the altitude drawn, by two straight cuts, into four parts, from which you can fold a square.

2016 Czech-Polish-Slovak Junior Match, 4

Tags: combinatorial geometry , tiling , combinatorics

Several tiles congruent to the one shown in the picture below are to be fit inside a $11 \times 11$ square table, with each tile covering $6$ whole unit squares, no sticking out the square and no overlapping. (a) Determine the greatest number of tiles which can be placed this way. (b) Find, with a proof, all unit squares which have to be covered in any tiling with the maximal number of tiles. [img]https://cdn.artofproblemsolving.com/attachments/c/d/23d93e9d05eab94925fc54006fe05123f0dba9.png[/img] Poland

2004 Denmark MO - Mohr Contest, 5

Tags: combinatorics , combinatorial geometry , tiling , tiles

Determine for which natural numbers $n$ you can cover a $2n \times 2n$ chessboard with non-overlapping $L$ pieces. An $L$ piece covers four spaces and has appearance like the letter $L$. The piece may be rotated and mirrored at will.

2020 Estonia Team Selection Test, 2

Tags: combinatorial geometry , angle , min , combinatorics , geometry

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

2005 Sharygin Geometry Olympiad, 24

Tags: combinatorial geometry , 3d geometry , angle , geometry

A triangle is given, all the angles of which are smaller than $\phi$, where $\phi <2\pi / 3$. Prove that in space there is a point from which all sides of the triangle are visible at an angle $\phi$.

2009 Mathcenter Contest, 3

Tags: combinatorial geometry , combinatorics , point

Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear. [i](Anonymous314)[/i] PS. wording needs to be fixed , [url=http://www.mathcenter.net/forum/showthread.php?t=7288]source[/url]

IV Soros Olympiad 1997 - 98 (Russia), 10.5

Tags: combinatorial geometry , combinatorics

In the lower left corner of the square $7 \times 7$ board there is a king. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different ways can the king get to the upper right corner of the board if he is prohibited from visiting the central square?

1997 May Olympiad, 5

Tags: combinatorial geometry , combinatorics

When Pablo turns $15$, he throws a party inviting $43$ friends. He presents them with a cake n the form of a regular $15$-sided polygon and on it $15$ candles. The candles are arranged so that between candles and vertices there are never three aligned (any three candles are not aligned, nor are any two candles with a vertex of the polygon, nor are any two vertices of the polygon with a candle). Then Pablo divides the cake into triangular pieces, by means of cuts that join candles to each other or candles and vertices, but also do not intersect with others already made. Why, by doing this, Paul was able to distribute a piece to each of his guests but he was left without eating?

2019 Czech-Polish-Slovak Junior Match, 5

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

Let $A_1A_2 ...A_{360}$ be a regular $360$-gon with centre $S$. For each of the triangles $A_1A_{50}A_{68}$ and $A_1A_{50}A_{69}$ determine, whether its images under some $120$ rotations with centre $S$ can have (as triangles) all the $360$ points $A_1, A_2, ..., A_{360}$ as vertices.

2007 Junior Tuymaada Olympiad, 3

Tags: divisible , combinatorial geometry , number theory , combinatorics

A square $ 600 \times 600$ divided into figures of $4$ cells of the forms in the figure: In the figures of the first two types in shaded cells The number $ 2 ^ k $ is written, where $ k $ is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by $9$.

ICMC 8, 6

Tags: combinatorial geometry

A set of points in the plane is called rigid if each point is equidistant from the three (or more) points nearest to it. (a) Does there exist a rigid set of $9$ points? (b) Does there exist a rigid set of $11$ points?

1958 Poland - Second Round, 2

Tags: combinatorial geometry , geometry , combinatorics

Six equal disks are placed on a plane so that their centers lie at the vertices of a regular hexagon with sides equal to the diameter of the disks. How many revolutions will a seventh disk of the same size make when rolling in the same plane externally over the disks before returning to its initial position?

1982 Brazil National Olympiad, 3

Tags: lattice points , combinatorics , combinatorial geometry

$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$?

1973 Polish MO Finals, 3

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

A polyhedron $W$ has the following properties: (i) It possesses a center of symmetry. (ii) The section of $W$ by a plane passing through the center of symmetry and one of its edges is always a parallelogram. (iii) There is a vertex of $W$ at which exactly three edges meet. Prove that $W$ is a parallelepiped.

1999 German National Olympiad, 6a

Tags: combinatorics , combinatorial geometry , isosceles right triangle

Suppose that an isosceles right-angled triangle is divided into $m$ acute-angled triangles. Find the smallest possible $m$ for which this is possible.