Olympiad problems

2010 Belarus Team Selection Test, 6.3

A $50 \times 50$ square board is tiled by the tetrominoes of the following three types: [img]https://cdn.artofproblemsolving.com/attachments/2/9/62c0bce6356ea3edd8a2ebfe0269559b7527f1.png[/img] Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling. (Folklore)

2011 Kurschak Competition, 3

Tags: combinatorial geometry , inequalities

Given $2n$ points and $3n$ lines on the plane. Prove that there is a point $P$ on the plane such that the sum of the distances of $P$ to the $3n$ lines is less than the sum of the distances of $P$ to the $2n$ points.

2024 India IMOTC, 1

Tags: combinatorics , combinatorial geometry

A sleeping rabbit lies in the interior of a convex $2024$-gon. A hunter picks three vertices of the polygon and he lays a trap which covers the interior and the boundary of the triangular region determined by them. Determine the minimum number of times he needs to do this to guarantee that the rabbit will be trapped. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]

2018 Hanoi Open Mathematics Competitions, 5

Tags: combinatorial geometry , combinatorics , coloring

The center of a circle and nine randomly selected points on this circle are colored in red. Every pair of those points is connected by a line segment, and every point of intersection of two line segments inside the circle is colored in red. What is the largest possible number of red points? A. $235$ B. $245$ C. $250$ D. $220$ E. $265$

2015 QEDMO 14th, 11

Tags: combinatorics , combinatorial geometry , chessboard

Let $m, n$ be natural numbers and let $m\cdot n$ be a multiple of $4$. A chessboard with $m \times n$ fields are covered with $1 \times 2$ large dominoes without gaps and without overlapping. Show that the number of dominoes that are parallel to a edge of the chess board is fixed . [hide=original wording] Seien m, n natu¨rliche Zahlen und sei m · n ein Vielfaches von 4. Ein Schachbrett mit m × n Feldern sei mit 1 × 2 großen Dominosteinen lu¨ckenlos und u¨berlappungsfrei u¨berdeckt. Zeige, dass die Anzahl der Dominosteine, die zu einer fest gew¨ahlten Kante des Schachbrettes parallel sind, gerade ist. [/hide]

BIMO 2022, 2

Tags: combinatorial geometry , combinatorics

Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $P$. What is the maximum possible value of $k$? [i]Proposed by Ivan Chan Kai Chin[/i]

Mathley 2014-15, 3

Tags: combinatorics , combinatorial geometry , isosceles

Given a regular $2013$-sided polygon, how many isosceles triangles are there whose vertices are vertices vertex of given polygon and haave an angle greater than $120^o$? Nguyen Tien Lam, High School for Natural Science,Hanoi National University.

1972 IMO Shortlist, 4

Tags: combinatorics , partition , point set , geometry , combinatorial geometry

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

2018 Costa Rica - Final Round, 1

Tags: combinatorial geometry , combinatorics , geometry

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).

1979 All Soviet Union Mathematical Olympiad, 277

Tags: square , geometry , combinatorial geometry , area

Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.

1972 All Soviet Union Mathematical Olympiad, 164

Tags: geometry , combinatorial geometry , area , square

Given several squares with the total area $1$. Prove that you can pose them in the square of the area $2$ without any intersections.

2009 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorial geometry , combinatorics , square

To obtain a square $P$ of side length $2$ cm divided into $4$ unit squares it is sufficient to draw $3$ squares: $P$ and another $2$ unit squares with a common vertex, as shown below: [img]https://cdn.artofproblemsolving.com/attachments/1/d/827516518871ec8ff00a66424f06fda9812193.png[/img] Find the minimum number of squares sufficient to obtain a square.of side length $n$ cm divided into $n^2$ unit squares ($n \ge 3$ is an integer).

1938 Moscow Mathematical Olympiad, 038

Tags: equidistant , 3d geometry , combinatorial geometry , geometry , plane

In space $4$ points are given. How many planes equidistant from these points are there? Consider separately (a) the generic case (the points given do not lie on a single plane) and (b) the degenerate cases.

2016 Saint Petersburg Mathematical Olympiad, 2

Tags: combinatorics , combinatorial geometry , square , chessboard , chess rook , rook

On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?

2013 Romania National Olympiad, 2

Tags: combinatorics , combinatorial geometry , chessboard

A rook starts moving on an infinite chessboard, alternating horizontal and vertical moves. The length of the first move is one square, of the second – two squares, of the third – three squares and so on. a) Is it possible for the rook to arrive at its starting point after exactly $2013$ moves? b) Find all $n$ for which it possible for the rook to come back to its starting point after exactly $n$ moves.

2003 May Olympiad, 4

Tags: geometry , combinatorics , combinatorial geometry , area

Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$. Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.

1989 All Soviet Union Mathematical Olympiad, 493

Tags: combinatorics , combinatorial geometry

One bird lives in each of $n$ bird-nests in a forest. The birds change nests, so that after the change there is again one bird in each nest. Also for any birds $A, B, C, D$ (not necessarily distinct), if the distance $AB < CD$ before the change, then $AB > CD$ after the change. Find all possible values of $n$.

1991 Tournament Of Towns, (316) 2

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

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)

1984 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , combinatorial geometry , geometry , geometric inequality

In a square field of side length $12$ there is a source that contains a system of straight irrigation ditches. This is laid out in such a way that for every point of the field the distance to the next ditch is at most $1$. Here, the source is as a point and are the ditches to be regarded as stretches. It must be verified that the total length of the irrigation ditches is greater than $70$ m. The sketch shows an example of a trench system of the type indicated. [img]https://cdn.artofproblemsolving.com/attachments/6/5/5b51511da468cf14b5823c6acda3c4d2fe8280.png[/img]

1969 IMO Shortlist, 68

Tags: geometry , point set , convex quadrilateral , combinatorial geometry

$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.

1996 Tournament Of Towns, (505) 2

Tags: trapezoid , equilateral , tiles , geometry , combinatorics , combinatorial geometry

For what positive integers $n$ is it possible to tile an equilateral triangle of side $n$ with trapezoids each of which has sides $1, 1, 1, 2$? (NB Vassiliev)

2024 IMC, 5

Tags: combinatorial geometry , probability

Let $n>d$ be positive integers. Choose $n$ independent, uniformly distributed random points $x_1,\dots,x_n$ in the unit ball $B \subset \mathbb{R}^d$ centered at the origin. For a point $p \in B$ denote by $f(p)$ the probability that the convex hull of $x_1,\dots,x_n$ contains $p$. Prove that if $p,q \in B$ and the distance of $p$ from the origin is smaller than the distance of $q$ from the origin, then $f(p) \ge f(q)$.

2025 Israel National Olympiad (Gillis), P4

Tags: geometry , rectangle , combinatorial geometry , combinatorics

A $100\times \sqrt{3}$ rectangular table is given. What is the minimum number of disk-shaped napkins of radius $1$ required to cover the table completely? [i]Remark:[/i] The napkins are allowed to overlap and protrude the table's edges.

1971 Swedish Mathematical Competition, 2

Tags: coloring , combinatorics , combinatorial geometry

An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.

2008 Germany Team Selection Test, 3

Tags: combinatorics , combinatorial geometry , polygon , extremal combinatorics

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]