Olympiad problems

2020 Chile National Olympiad, 2

The points of this lattice $4\times 4 = 16$ points can be vertices of squares. [asy] unitsize(1 cm); int i, j; for (i = 0; i <= 3; ++i) { draw((i,0)--(i,3)); draw((0,i)--(3,i)); } draw((1,1)--(2,2)--(1,3)--(0,2)--cycle); for (i = 0; i <= 3; ++i) { for (j = 0; j <= 3; ++j) { dot((i,j)); }} [/asy] Calculate the number of different squares that can be formed in a lattice of $100\times 100$ points.

1984 Tournament Of Towns, (072) 3

Tags: point , segment , combinatorial geometry , geometry

On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely? (V.E. Kolosov)

2018 Latvia Baltic Way TST, P7

Tags: combinatorics , combinatorial geometry

Let $n \ge 3$ points be given in the plane, no three of which lie on the same line. Determine whether it is always possible to draw an $n$-gon whose vertices are the given points and whose sides do not intersect. [i]Remark.[/i] The $n$-gon can be concave.

2002 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: game strategy , combinatorial geometry , minimum , combinatorics

Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.

2023 Ukraine National Mathematical Olympiad, 11.2

Tags: combinatorial geometry , combinatorics

Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane so that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What largest possible number of these angles can be equal to $90^\circ$? [i]Proposed by Anton Trygub[/i]

1991 Tournament Of Towns, (316) 2

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

Is it possible to divide the plane into polygons so that each polygon is transformed into itself under some rotation by $360/7$ degrees about some point? All sides of these polygons must be greater than $1$ cm. (A polygon is the part of a plane bounded by one non-self-intersect-ing closed broken line, not necessarily convex.) (A. Andjans, Riga)

I Soros Olympiad 1994-95 (Rus + Ukr), 9.2

Tags: strategy , regular , equilateral , combinatorics , combinatorial geometry , game

Given a regular $72$-gon. Lenya and Kostya play the game "Make an equilateral triangle." They take turns marking with a pencil on one still unmarked angle of the $72$-gon: Lenya uses red. Kostya uses blue. Lenya starts the game, and the one who marks first wins if its color is three vertices that are the vertices of some equilateral triangle, if all the vertices are marked and no such a triangle exists, the game ends in a draw. Prove that Kostya can play like this so as not to lose.

III Soros Olympiad 1996 - 97 (Russia), 11.5

Tags: geometry , combinatorial geometry

Prove that this triangle cut out of paper can be folded so that the surface of a regular unit tetradragon (i.e., a triangular pyramid, all edges of which are equal to $1$) is obtained if: a) this triangle is isosceles, the lateral sides are equal to $2$ , the angle between them is $120^o$, b) two sides of this triangle are equal to $2$ and $2\sqrt3$, the angle between them is $150^o$.

2015 Romanian Master of Mathematics, 2

Tags: combinatorial geometry , combinatorics

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

1999 Cono Sur Olympiad, 5

Tags: distance , square , combinatorial geometry , combinatorics

Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.

2003 Junior Balkan Team Selection Tests - Moldova, 8

Tags: combinatorial geometry , sum , combinatorics

In the rectangular coordinate system every point with integer coordinates is called laticeal point. Let $P_n(n, n + 5)$ be a laticeal point and denote by $f(n)$ the number of laticeal points on the open segment $(OP_n)$, where the point $0(0,0)$ is the coordinates system origine. Calculate the number $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.

1994 All-Russian Olympiad, 4

Tags: combinatorics , combinatorial geometry

On a line are given $n$ blue and $n$ red points. Prove that the sum of distances between pairs of points of the same color does not exceed the sum of distances between pairs of points of different colors. (O. Musin)

2023 Kyiv City MO Round 1, Problem 5

Tags: combinatorics , square grid , combinatorial geometry

You are given a square $n \times n$. The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$, find the largest value of $m$ for which it is possible. [i]Proposed by Oleksiy Masalitin, Fedir Yudin[/i]

1997 USAMO, 4

Tags: geometry , geometric transformation , combinatorial geometry , modular arithmetic , homothety

To [i]clip[/i] a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$. Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$, and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$, for all $n \geq 6$.

Cono Sur Shortlist - geometry, 2009.G1.6

Tags: rectangle , geometry , combinatorial geometry

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

2000 Tournament Of Towns, 4

Tags: convex polygon , lattice points , sum , combinatorics , combinatorial geometry

Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form $x = m$, $m$ an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form $y = n$, $n$ an integer, that lie within the polygon. (G Galperin)

Kvant 2022, M2708 a)

Tags: combinatorics , combinatorial geometry , lattice points

Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?

2001 Nordic, 1

Tags: combinatorics , coordinates , combinatorial geometry , integer , analytic geometry

Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.

2021 China Team Selection Test, 5

Tags: combinatorics , combinatorial geometry , area , convex hull , convex polygon , symmetry , geometry

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

1990 Austrian-Polish Competition, 8

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

We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

2013 Tournament of Towns, 5

Tags: geometry , combinatorial geometry , parallel , lattice points

A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.

2012 Tournament of Towns, 2

Tags: point , combinatorial geometry , combinatorics

One hundred points are marked in the plane, with no three in a line. Is it always possible to connect the points in pairs such that all fifty segments intersect one another?

VI Soros Olympiad 1999 - 2000 (Russia), 10.2

Tags: combinatorial geometry , point , geometry , combinatorics

$37$ points are arbitrarily marked on the plane. Prove that among them there must be either two points at a distance greater than $6$, or two points at a distance less than $1.5$.

2025 Sharygin Geometry Olympiad, 23

Tags: combinatorial geometry , geometry

Let us say that a subset $M$ of the plane contains a hole if there exists a disc not contained in $M$, but contained inside some polygon with the boundary lying in $M$. Can the plane be presented as a union of $n$ convex sets such that the union of any $n-1$ from them contains a hole? Proposed by: N.Spivak

2011 QEDMO 8th, 6

Tags: combinatorial geometry , combinatorics , geometry

A [i]synogon [/i] is a convex $2n$-gon with all sides of the same length and all opposite sides are parallel. Show that every synogon can be broken down into a finite number of rhombuses.