Olympiad problems

1979 All Soviet Union Mathematical Olympiad, 277

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

1972 Dutch Mathematical Olympiad, 4

Tags: combinatorics , combinatorial geometry

On a circle with radius $1$ the points $A_1, A_2,..., A_n$ lie such that every arc $A_iA_{i+i}$ has length $\frac{2\pi}{n}= a$. Given that there exists a set $V$ consisting of $ k$ of these points ($k < n$), which has the property that each of the arc lengths $a$, $2a$$,...$, $(n- 1)a$ can be obtained in exactly one way be taken as the length of an arc traversed in a positive sense, beginning and ending in a point of $V$. Express $n$ in terms of $k$ and give the set $V$ for the case $n = 7$.

1955 Moscow Mathematical Olympiad, 295

Tags: convex , line , geometry , combinatorial geometry

Which convex domains (figures) on a plane can contain an entire straight line? It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

1982 Tournament Of Towns, (031) 5

Tags: combinatorics , combinatorial geometry

The plan of a Martian underground is represented by a closed selfintersecting curve, with at most one self-intersection at each point. Prove that a tunnel for such a plan may be constructed in such a way that the train passes consecutively over and under the intersecting parts of the tunnel.

1976 Chisinau City MO, 127

Tags: combinatorics , combinatorial geometry

The convex $1976$-gon is divided into $1975$ triangles. Prove that there is a straight line separating one of these triangles from the rest.

2008 Swedish Mathematical Competition, 4

Tags: angle , geometry , combinatorial geometry

A convex $n$-side polygon has angles $v_1,v_2,\dots,v_n$ (in degrees), where all $v_k$ ($k = 1,2,\dots,n$) are positive integers divisible by $36$. (a) Determine the largest $n$ for which this is possible. (b) Show that if $n>5$, two of the sides of the $n$-polygon must be parallel.

2018 Hanoi Open Mathematics Competitions, 15

Tags: combinatorial geometry , combinatorics

There are $n$ distinct straight lines on a plane such that every line intersects exactly $12$ others. Determine all the possible values of $n$.

2004 Estonia Team Selection Test, 6

Tags: polyhedron , pentagon , 3d geometry , hexagon , geometry , combinatorial geometry , combinatorics

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

2019 Canadian Mathematical Olympiad Qualification, 2

Tags: combinatorial geometry , pyramid , equilateral , sphere , combinatorics

Rosemonde is stacking spheres to make pyramids. She constructs two types of pyramids $S_n$ and $T_n$. The pyramid $S_n$ has $n$ layers, where the top layer is a single sphere and the $i^{th}$ layer is an $i\times $i square grid of spheres for each $2 \le i \le n$. Similarly, the pyramid $T_n$ has $n$ layers where the top layer is a single sphere and the $i^{th}$ layer is $\frac{i(i+1)}{2}$ spheres arranged into an equilateral triangle for each $2 \le i \le n$.

2022 Grosman Mathematical Olympiad, P3

Tags: combinatorial geometry , combinatorics

An ant crawled a total distance of $1$ in the plane and returned to its original position (so that its path is a closed loop of length $1$; the width is considered to be $0$). Prove that there is a circle of radius $\frac{1}{4}$ containing the path. Illustration of an example path:

2008 Germany Team Selection Test, 3

Tags: combinatorics , combinatorial geometry , extremal combinatorics , polygon

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]

2019 Estonia Team Selection Test, 11

Tags: combinatorial geometry , perimeter , geometry , circumcircle

Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply: (a) the circumcircle of each triangle in the set $T$ is $\omega$; (b) The interior of any two triangles in the set $T$ has no common point. Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.

2009 BAMO, 1

Tags: combinatorial geometry , combinatorics

A square grid of $16$ dots (see the figure) contains the corners of nine $1\times1$ squares, four $2\times 2$ squares, and one $3\times3$ square, for a total of $14$ squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the $14$ squares is missing at least one corner? Justify your answer by showing both that the number of dots you claim is sufficient and by explaining why no smaller number of dots will work. [img]https://cdn.artofproblemsolving.com/attachments/0/9/bf091a769dbec40eceb655f5588f843d4941d6.png[/img]

1998 Mexico National Olympiad, 6

Tags: geometry , combinatorial geometry , 3-dimensional geometry , distance

A plane in space is equidistant from a set of points if its distances from the points in the set are equal. What is the largest possible number of equidistant planes from five points, no four of which are coplanar?

2015 China Northern MO, 6

Tags: combinatorial geometry , tiling , tiles , combinatorics

The figure obtained by removing one small unit square from the $2\times 2$ grid table is called an $L$ ''shape". .Put $k$ L-shapes in an $8\times 8$ grid table. Each $L$-shape can be rotated, but each $L$ shape is required to cover exactly three small unit squares in the grid table, and the common area covered by any two $L$ shapes is $0$, and except for these $k$ $L$ shapes, no other $L$ shapes can be placed. Find the minimum value of $k$.

1975 Bulgaria National Olympiad, Problem 6

Tags: combinatorics , geometry , combinatorial geometry

Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$. [i](H. Lesov)[/i]

2018 Oral Moscow Geometry Olympiad, 6

Tags: equilateral triangle , equilateral , geometry , combinatorial geometry

Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.

2019 Tournament Of Towns, 3

Tags: bicentric quadrilateral , tiling , cut , combinatorial geometry , combinatorics

Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed. (Nairi Sedrakyan)

1985 All Soviet Union Mathematical Olympiad, 399

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

Given a straight line $\ell$ and the point $O$ out of the line. Prove that it is possible to move an arbitrary point $A$ in the same plane to the $O$ point, using only rotations around $O$ and symmetry with respect to the $\ell$.

2012 IMAR Test, 4

Tags: combinatorial geometry , coloring , geometry

Design a planar finite non-empty set $S$ satisfying the following two conditions: (a) every line meets $S$ in at most four points; and (b) every $2$-colouring of $S$ - that is, each point of $S$ is coloured one of two colours - yields (at least) three monochromatic collinear points.

1999 Mexico National Olympiad, 4

Tags: combinatorics , board , combinatorial geometry , distance

An $8 \times 8$ board is divided into unit squares. Ten of these squares have their centers marked. Prove that either there exist two marked points on the distance at most $\sqrt2$, or there is a point on the distance $1/2$ from the edge of the board.

2008 Brazil National Olympiad, 2

Tags: combinatorics , combinatorial geometry

Let $ S$ be a set of $ 6n$ points in a line. Choose randomly $ 4n$ of these points and paint them blue; the other $ 2n$ points are painted green. Prove that there exists a line segment that contains exactly $ 3n$ points from $ S$, $ 2n$ of them blue and $ n$ of them green.

2011 Tournament of Towns, 4

Tags: coloring , polygon , combinatorial geometry , combinatorics

There are $n$ red sticks and $n$ blue sticks. The sticks of each colour have the same total length, and can be used to construct an $n$-gon. We wish to repaint one stick of each colour in the other colour so that the sticks of each colour can still be used to construct an $n$-gon. Is this always possible if (a) $n = 3$, (b) $n > 3$ ?

2024 Miklos Schweitzer, 8

Tags: graph theory , planar graph , combinatorial geometry

Prove that for any finite bipartite planar graph, a circle can be assigned to each vertex so that all circles are coplanar, the circles assigned to any two adjacent vertices are tangent to one another, while the circles assigned to any two distinct, non-adjacent vertices intersect in two points.

2009 Canada National Olympiad, 5

Tags: geometry , circumcircle , combinatorial geometry

A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius $1$. Prove that the set of all marked points can be covered with a disk of radius $1$.