Olympiad problems

2023 Peru MO (ONEM), 3

Prove that, for every integer $n \ge 2$, it is possible to divide a regular hexagon into $n$ quadrilaterals such that any two of them are similar. Clarification: Two quadrilaterals are similar if they have their corresponding sides proportional and their corresponding angles are equal, that is, the quadrilaterals $ABCD$ and $EFGH$ are similar if $\frac{AB}{EF}= \frac{BC}{FG}= \frac{CD}{GH} = \frac{DA}{HE}$, $\angle ABC = \angle EFG$, $\angle BCD = \angle FGH$, $\angle CDA = \angle GHE$ and $\angle DAB = \angle HEF$.

2012 Denmark MO - Mohr Contest, 2

Tags: square , geometry , rectangle , combinatorial geometry

It is known about a given rectangle that it can be divided into nine squares which are situated relative to each other as shown. The black rectangle has side length $1$. Are there more than one possibility for the side lengths of the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/0/af6bc5b867541c04586e4b03db0a7f97f8fe87.png[/img]

1973 Kurschak Competition, 2

Tags: combinatorial geometry , lattice , combinatorics

For any positive real $r$, let $d(r)$ be the distance of the nearest lattice point from the circle center the origin and radius $r$. Show that $d(r)$ tends to zero as $r$ tends to infinity.

2019 Iran Team Selection Test, 5

Tags: combinatorial geometry , combinatorics , perimeter

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]

1978 Chisinau City MO, 163

Tags: point , polyline , geometry , combinatorial geometry , geometric inequality

On the plane $n$ points are selected that do not belong to one straight line. Prove that the shortest closed path passing through all these points is a non-self-intersecting polygon.

2007 BAMO, 2

Tags: combinatorial geometry , combinatorics , parallelogram , coloring

The points of the plane are colored in black and white so that whenever three vertices of a parallelogram are the same color, the fourth vertex is that color, too. Prove that all the points of the plane are the same color.

1957 Moscow Mathematical Olympiad, 358

Tags: combinatorial geometry , broken line , 3d geometry , geometry , perpendicular

The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by $6$.

2002 Switzerland Team Selection Test, 1

Tags: point , plane , combinatorics , combinatorial geometry

In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.

2019 Istmo Centroamericano MO, 5

Tags: combinatorial geometry , combinatorics , regular polygon , triangle

Gabriel plays to draw triangles using the vertices of a regular polygon with $2019$ sides, following these rules: (i) The vertices used by each triangle must not have been previously used. (ii) The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn. If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.

1999 May Olympiad, 4

Tags: combinatorics , combinatorial geometry

Ten square cardboards of $3$ centimeters on a side are cut by a line, as indicated in the figure. After the cuts, there are $20$ pieces: $10$ triangles and $10$ trapezoids. Assemble a square that uses all $20$ pieces without overlaps or gaps. [img]https://cdn.artofproblemsolving.com/attachments/7/9/ec2242cca617305b02eef7a5409e6a6b482d66.gif[/img]

2002 All-Russian Olympiad Regional Round, 9.4

Tags: combinatorics , combinatorial geometry , rectangle , geometry

Located on the plane $\left[ \frac43 n \right]$ rectangles with sides parallel to the coordinate axes. It is known that any rectangle intersects at least n rectangles. Prove that exists a rectangle that intersects all rectangles.

1983 Tournament Of Towns, (042) O5

Tags: combinatorial geometry , combinatorics , projection , regular polygon

A point is chosen inside a regular $k$-gon in such a way that its orthogonal projections on to the sides all meet the respective sides at interior points. These points divide the sides into $2k$ segments. Let these segments be enumerated consecutively by the numbers $1,2, 3, ... ,2k$. Prove that the sum of the lengths of the segments having even numbers equals the sum of the segments having odd numbers. (A Andjans, Riga)

1993 Tournament Of Towns, (368) 7

Tags: combinatorics , combinatorial geometry , coloring

Two coloured points are marked on a line, with the blue one at the left and the red one at the right. You may add to the line two neighbouring points of the same color (both red or both blue) or delete two such points (neighbouring means that there is no coloured point between these two). Prove that after several such transformation you cannot again get only two points on the line in which the red one is at the left and the blue one is at the right. (A Belov)

2015 Estonia Team Selection Test, 8

Tags: diagonal , combinatorial geometry , combinatorics , regular polygon

Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.

2003 Romania National Olympiad, 4

Tags: function , combinatorial geometry

Let $ P$ be a plane. Prove that there exists no function $ f: P\rightarrow P$ such that for every convex quadrilateral $ ABCD$, the points $ f(A),f(B),f(C),f(D)$ are the vertices of a concave quadrilateral. [i]Dinu Şerbănescu[/i]

1993 Tournament Of Towns, (361) 4

Tags: combinatorics , cube , combinatorial geometry , geometry , 3d geometry

An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times? (S Tokarev)

1993 Bundeswettbewerb Mathematik, 1

Tags: combinatorics , combinatorial geometry , coloring

In a regular nonagon, each vertex is colored either red or green. Three corners of the nonagon determine a triangle. Such a triangle is called [i]red [/i] or [i]green [/i] if all its vertices are red or green if all are green. Prove that for each such coloring of the nonagon there are at least two different ones , that are congruent triangles of the same color.

1965 IMO, 6

Tags: geometry , diameter , point set , combinatorial geometry

In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

2017 Hanoi Open Mathematics Competitions, 9

Tags: combinatorics , combinatorial geometry

Cut off a square carton by a straight line into two pieces, then cut one of two pieces into two small pieces by a straight line, ect. By cutting $2017$ times we obtain $2018$ pieces. We write number $2$ in every triangle, number 1 in every quadrilateral, and $0$ in the polygons. Is the sum of all inserted numbers always greater than $2017$?

1956 Moscow Mathematical Olympiad, 341

Tags: geometry , 3d geometry , combinatorial geometry , cube , combinatorics

$1956$ points are chosen in a cube with edge $13$. Is it possible to fit inside the cube a cube with edge $1$ that would not contain any of the selected points?

1991 Tournament Of Towns, (316) 2

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

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)

1989 Balkan MO, 2

Tags: polynomial , gauss , calculus , derivative , combinatorial geometry , absolute value , algebra

Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.

1981 All Soviet Union Mathematical Olympiad, 324

Tags: distance , combinatorial geometry , geometry , rectangle

Six points are marked inside the $3\times 4$ rectangle. Prove that there is a pair of marked points with the distance between them not greater than $\sqrt5$.

2024 China Team Selection Test, 6

Tags: combinatorics , combinatorial geometry

Let $m,n>2$ be integers. A regular ${n}$-sided polygon region $\mathcal T$ on a plane contains a regular ${m}$-sided polygon region with a side length of ${}{}{}1$. Prove that any regular ${m}$-sided polygon region $\mathcal S$ on the plane with side length $\cos{\pi}/[m,n]$ can be translated inside $\mathcal T.$ In other words, there exists a vector $\vec\alpha,$ such that for each point in $\mathcal S,$ after translating the vector $\vec\alpha$ at that point, it fall into $\mathcal T.$ Note: The polygonal area includes both the interior and boundaries. [i]Created by Bin Wang[/i]

1983 All Soviet Union Mathematical Olympiad, 362

Tags: square , infinite board , sum , combinatorial geometry , combinatorics

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?