Olympiad problems

2011 Philippine MO, 3

The $2011$th prime number is $17483$ and the next prime is $17489$. Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?

1992 Tournament Of Towns, (323) 4

Tags: combinatorics , combinatorial geometry

A circle is divided into $7$ arcs. The sum of the angles subtending any two neighbouring arcs is no more than $103^o$. Find the maximal number $A$ such that any of the $7$ arcs is subtended by no less than $A^o$. Prove that this value $A$ is really maximal. (A. Tolpygo, Kiev)

2001 All-Russian Olympiad Regional Round, 10.8

Tags: combinatorial geometry , combinatorics

There are a thousand non-intersecting arcs on a circle, and on each of them contains two natural numbers. Sum of numbers of each arc is divided by the product of the numbers of the arc following it clockwise arrow. What is the largest possible value of the largest number written?

1999 Poland - Second Round, 2

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

A cube of edge $2$ with one of the corner unit cubes removed is called a [i]piece[/i]. Prove that if a cube $T$ of edge $2^n$ is divided into $2^{3n}$ unit cubes and one of the unit cubes is removed, then the rest can be cut into [i]pieces[/i].

2016 Romania National Olympiad, 1

Tags: combinatorial geometry , combinatorics , 3d geometry , prism , coloring , geometry

The vertices of a prism are colored using two colors, so that each lateral edge has its vertices differently colored. Consider all the segments that join vertices of the prism and are not lateral edges. Prove that the number of such segments with endpoints differently colored is equal to the number of such segments with endpoints of the same color.

1985 IMO Longlists, 84

Tags: combinatorial geometry , combinatorics , graph theory , extremal combinatorics , extremal graph theory

Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.

2016 Hanoi Open Mathematics Competitions, 7

Tags: combinatorial geometry , combinatorics , grid

Nine points form a grid of size $3\times 3$. How many triangles are there with $3$ vertices at these points?

2008 Thailand Mathematical Olympiad, 10

Tags: combinatorial geometry , combinatorics , point

On the sides of triangle $\vartriangle ABC$, $17$ points are added, so that there are $20$ points in total (including the vertices of $\vartriangle ABC$.) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.

2015 Swedish Mathematical Competition, 5

Tags: combinatorial geometry , geometry , combinatorics

Given a finite number of points in the plane as well as many different rays starting at the origin. It is always possible to pair the points with the rays so that they parallell displaced rays starting in respective points do not intersect?

2017 Canada National Olympiad, 5

Tags: combinatorial geometry , discrete geometry

There are $100$ circles of radius one in the plane. A triangle formed by the centres of any three given circles has area at most $2017$. Prove that there is a line intersecting at least three of the circles.

2021 Israel TST, 2

Tags: combinatorial geometry , combinatorics , distance

Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$. Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.

1982 Czech and Slovak Olympiad III A, 4

Tags: geometry , combinatorics , point , combinatorial geometry

In a circle with a radius of $1$, $64$ mutually different points are selected. Prove that $10$ mutually different points can be selected from them, which lie in a circle with a radius $\frac12$.

2023 Portugal MO, 3

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

A crate with a base of $4 \times 2$ and a height of $2$ is open at the top. Tomas wants to completely fill the crate with some of his cubes. It has $16$ equal cubes of volume $1$ and two equal cubes of volume $8$. A cube of volume $1$ can only be placed on the top layer if the cube on the bottom layer has already been placed. In how many ways can Tom'as fill the box with cubes, placing them one by one?

2011 Tournament of Towns, 2

Tags: combinatorial geometry , analytic geometry , rectangle , geometry , parallel , lattice points

In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is $2011$, prove that the sides of the rectangular box are parallel to the coordinate axes.

2019 China Team Selection Test, 3

Tags: geometry , combinatorics , combinatorial geometry

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

II Soros Olympiad 1995 - 96 (Russia), 11.5

Tags: combinatorial geometry , combinatorics

The space is filled in the usual way with unit cubes. (Each cube is adjacent to $6$ others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of $1/19$, $1/9$ and $1/7$ from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?

2022 Mexico National Olympiad, 6

Tags: geometry , combinatorial geometry , similar triangles

Find all integers $n\geq 3$ such that there exists a convex $n$-gon $A_1A_2\dots A_n$ which satisfies the following conditions: - All interior angles of the polygon are equal - Not all sides of the polygon are equal - There exists a triangle $T$ and a point $O$ inside the polygon such that the $n$ triangles $OA_1A_2,\ OA_2A_3,\ \dots,\ OA_{n-1}A_n,\ OA_nA_1$ are all similar to $T$, not necessarily in the same vertex order.

2023 China Team Selection Test, P12

Tags: combinatorial geometry , combinatorics , analytic geometry , lattice

Prove that there exists some positive real number $\lambda$ such that for any $D_{>1}\in\mathbb{R}$, one can always find an acute triangle $\triangle ABC$ in the Cartesian plane such that [list] [*] $A, B, C$ lie on lattice points; [*] $AB, BC, CA>D$; [*] $S_{\triangle ABC}<\frac{\sqrt 3}{4}D^2+\lambda\cdot D^{4/5}$.

2014 Chile National Olympiad, 5

Tags: combinatorics , combinatorial geometry , parallelogram , geometry

Prove that if a quadrilateral $ABCD$ can be cut into a finite number of parallelograms, then $ABCD$ is a parallelogram.

1973 Chisinau City MO, 65

Tags: geometric inequality , combinatorial geometry , geometry , chord , combinatorics

A finite number of chords is drawn in a circle $1$ cm in diameter so that any diameter of the circle intersects at most $N$ of these chords. Prove that the sum of the lengths of all chords is less than $3.15 \cdot N$ cm.

1996 Rioplatense Mathematical Olympiad, Level 3, 5

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

There is a board with $n$ rows and $4$ columns, and white, yellow and light blue chips. Player $A$ places four tokens on the first row of the board and covers them so Player $B$ doesn't know them. How should player $B$ do to fill the minimum number of rows with chips that will ensure that in any of the rows he will have at least three hits? Clarification: A hit by player $B$ occurs when he places a token of the same color and in the same column as $A$.

2001 German National Olympiad, 4

Tags: combinatorial geometry , combinatorics

In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn. [img]https://cdn.artofproblemsolving.com/attachments/0/5/33795820e0335686b06255180af698e536a9be.png[/img]

2019 Nigerian Senior MO Round 3, 4

Tags: combinatorial geometry , integer , grid , rectangle , combinatorics

A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings. Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]

1998 Israel National Olympiad, 7

Tags: geometry , polygon , parallel , combinatorial geometry , combinatorics

A polygonal line of the length $1001$ is given in a unit square. Prove that there exists a line parallel to one of the sides of the square that meets the polygonal line in at least $500$ points.

2003 All-Russian Olympiad Regional Round, 9.8

Tags: combinatorial geometry , geometry , combinatorics , convex , convex polygon , acute

Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.