Olympiad problems

2007 Moldova Team Selection Test, 4

Consider five points in the plane, no three collinear. The convex hull of this points has area $S$. Prove that there exist three points of them that form a triangle with area at most $\frac{5-\sqrt 5}{10}S$

2003 Bulgaria Team Selection Test, 3

Tags: combinatorial geometry , induction , combinatorics

Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.

2022 Francophone Mathematical Olympiad, 2

Tags: coloring , combinatorial geometry , combinatorics

We consider an $n \times n$ table, with $n\ge1$. Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish is achievable?

1987 Greece National Olympiad, 4

Tags: combinatorial geometry , combinatorics , geometry

Consider a convex $100$-gon $A_1A_2...A_{100}$. Draw the diagonal $A_{43}A_{81}$ which divides it into two convex polygons $P_1,P_2$. How many vertices and how diagonals, has each of the polygons $P_1,P_2$?

2013 QEDMO 13th or 12th, 5

Tags: chessboard , combinatorial geometry , combinatorics

$16$ pieces of the shape $1\times 3$ are placed on a $7\times 7$ chessboard, each of which is exactly three fields. One field remains free. Find all possible positions of this field.

2022 Caucasus Mathematical Olympiad, 4

Tags: lattice points , combinatorial geometry , combinatorics

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

2014 Argentina National Olympiad Level 2, 5

Tags: combinatorial geometry , combinatorics

Let $A{}$ be a point in the Cartesian plane. At each step, Ann tells Bob a number $0< a\leqslant 1$ and he then moves $A{}$ in one of the four cardinal directions, at his choice, by a distance of $a{}$. This process cotinues as long as Ann wishes. Amongst every $100$ consecutive moves, each of the four possible moves should have been made at least once. Ann's goal is to force Bob to eventually choose a point at a distance greater than $100$ from the initial position of $A{}$. Can Ann achieve her goal?

1988 Tournament Of Towns, (196) 3

Tags: polyhedron , combinatorial geometry , combinatorics

Prove that for each vertex of a polyhedron it is possible to attach a natural number so that for each pair of vertices with a common edge, the attached numbers are not relatively prime (i.e. they have common divisors), and with each pair of vertices without a common edge the attached numbers are relatively prime. (Note: there are infinitely many prime numbers.)

1972 Poland - Second Round, 2

Tags: combinatorial geometry , rectangle , combinatorics , geometry

In a rectangle with sides of length 20 and 25 there are 120 squares of side length 1. Prove that there is a circle with a diameter of 1 contained in this rectangle and having no points in common with any of these squares.

1987 Tournament Of Towns, (140) 5

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

A certain number of cubes are painted in six colours, each cube having six faces of different colours (the colours in different cubes may be arranged differently) . The cubes are placed on a table so as to form a rectangle. We are allowed to take out any column of cubes, rotate it (as a whole) along its long axis and replace it in the rectangle. A similar operation with rows is also allowed. Can we always make the rectangle monochromatic (i.e. such that the top faces of all the cubes are the same colour) by means of such operations? ( D. Fomin , Leningrad)

2017 Hanoi Open Mathematics Competitions, 15

Tags: geometry , square , combinatorial geometry

Let $S$ denote a square of side-length $7$, and let eight squares with side-length $3$ be given. Show that it is impossible to cover $S$ by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of $S$ .

1961 All-Soviet Union Olympiad, 2

Tags: combinatorial geometry , rectangle , geometry , circles

Consider $120$ unit squares arbitrarily situated in a $20\times 25$ rectangle. Prove that one can place a circle with unit diameter in the rectangle without intersecting any of the squares.

1997 Israel National Olympiad, 7

Tags: combinatorial geometry , combinatorics

A square with side $10^6$, with a corner square with side $10^{-3}$ cut off, is partitioned into $10$ rectangles. Prove that at least one of these rectangles has the ratio of the greater side to the smaller one at least $9$.

III Soros Olympiad 1996 - 97 (Russia), 11.6

Tags: combinatorial geometry , combinatorics , geometry

What is the largest number of obtuse triangles that can be composed of $16$ different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?

1995 Tournament Of Towns, (471) 5

Tags: combinatorial geometry , combinatorics , geometry

A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line. (a) Do there exist two congruent simple $7$-gons in the plane such that all the seven vertices of one of the $7$-gons are the vertices of the other one and yet these two $7$-gons have no common sides? (b) Do there exist three such $7$-gons? (V Proizvolov)

1990 All Soviet Union Mathematical Olympiad, 513

Tags: point , edge , graph , combinatorial geometry , combinatorics

A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.

1971 IMO Shortlist, 2

Tags: euclidean distance , point set , combinatorial geometry , geometry

Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

2014 Danube Mathematical Competition, 4

Tags: combinatorics , cardinality , lattice points , maximal , parallelogram , geometry , combinatorial geometry

Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.

1953 Moscow Mathematical Olympiad, 251

Tags: point , combinatorial geometry , convex polygon , circles

On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?

2021 Czech-Polish-Slovak Junior Match, 3

Tags: combinatorial geometry , combinatorics

A [i]cross [/i] is the figure composed of $6$ unit squares shown below (and any figure made of it by rotation). [img]https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png[/img] Find the greatest number of crosses that can be cut from a $6 \times 11$ divided sheet of paper into unit squares (in such a way that each cross consists of six such squares).

1986 Brazil National Olympiad, 3

Tags: combinatorial geometry , angle , geometry

The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.

2013 Thailand Mathematical Olympiad, 7

Tags: combinatorial geometry , combinatorics

Let $P_1, ... , P_{2556}$ be distinct points in a regular hexagon $ABCDEF$ with unit side length. Suppose that no three points in the set $S = \{A, B, C, D, E, F, P_1, ... , P_{2556}\}$ are collinear. Show that there is a triangle whose vertices are in $S$ and whose area is less than $\frac{1}{1700}$ .

2021 Kyiv Mathematical Festival, 1

Tags: combinatorial geometry , arithmetic sequence , geometry

Is it possible to mark four points on the plane so that the distances between any point and three other points form an arithmetic progression? (V. Brayman)

1999 All-Russian Olympiad Regional Round, 9.5

Tags: tiles , combinatorics , combinatorial geometry

All cells of the checkered plane are painted in $5$ colors so that in any figure of the species [img]https://cdn.artofproblemsolving.com/attachments/f/f/49b8d6db20a7e9cca7420e4b51112656e37e81.png[/img] all colors are different. Prove that in any figure of the species $ \begin{tabular}{ | l | c| c | c | r| } \hline & & & &\\ \hline \end{tabular}$, all colors are different..

1997 IMO Shortlist, 3

Tags: extremal combinatorics , geometry , vector , combinatorial geometry , combinatorics

For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$