Olympiad problems

1955 Moscow Mathematical Olympiad, 313

On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.

2021 Iran MO (2nd Round), 4

Tags: geometry , combinatorics , combinatorial geometry

$n$ points are given on a circle $\omega$. There is a circle with radius smaller than $\omega$ such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of $\omega$ with endpoints not belonging to the given points such that all the $n$ given points remain in one side of the diameter.

2021 Thailand TSTST, 2

Tags: combinatorics , geometry , combinatorial geometry

Let $n$ be a positive integer and let $0\leq k\leq n$ be an integer. Show that there exist $n$ points in the plane with no three on a line such that the points can be divided into two groups satisfying the following properties. $\text{(i)}$ The first group has $k$ points and the distance between any two distinct points in this group is irrational. $\text{(ii)}$ The second group has $n-k$ points and the distance between any two distinct points in this group is an integer. $\text{(iii)}$ The distance between a point in the first group and a point in the second group is irrational.

Kvant 2022, M2708 a)

Tags: combinatorial geometry , lattice points , combinatorics

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

2010 All-Russian Olympiad Regional Round, 10.8

Tags: combinatorics , combinatorial geometry

Let's call it a [i] staircase of height [/i]$n$, a figure consisting from all square cells $n\times n$ lying no higher diagonals (the figure shows a [i]staircase of height [/i] $4$ ). In how many different ways can a [i]staircase of height[/i] $n$ can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs? [img]https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png[/img]

1989 Tournament Of Towns, (216) 4

Tags: combinatorial geometry , geometry , 3d geometry

Is it possible to mark a diagonal on each little square on the surface of a Rubik 's cube so that one obtains a non-intersecting path? (S . Fomin, Leningrad)

2007 Finnish National High School Mathematics Competition, 3

Tags: combinatorial geometry , geometry

There are five points in the plane, no three of which are collinear. Show that some four of these points are the vertices of a convex quadrilateral.

1971 Bulgaria National Olympiad, Problem 3

Tags: combinatorial geometry , combinatorics , geometry

There are given $20$ points in the plane, no three of which lie on a single line. Prove that there exist at least $969$ quadrilaterals with vertices from the given points.

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: combinatorics , combinatorial geometry , equilateral

A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines.

Kvant 2020, M2591

Tags: combinatorics , combinatorial geometry

There are 100 blue lines drawn on the plane, among which there are no parallel lines and no three of which pass through one point. The intersection points of the blue lines are marked in red. Could it happen that the distance between any two red dots lying on the same blue line is equal to an integer? [i]From the folklore[/i]

2008 Indonesia TST, 1

Tags: geometry , combinatorial geometry , combinatorics , area of a triangle

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

2012 Bundeswettbewerb Mathematik, 4

Tags: combinatorial geometry , regular polygon , isosceles , trapezoid , isosceles triangle , combinatorics , geometry

From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.

2020 Ukrainian Geometry Olympiad - December, 2

Tags: combinatorics , combinatorial geometry , isosceles , geometry

On a straight line lie $100$ points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these $101$ points?

2019 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorial geometry , combinatorics , rectangle

In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.

1988 All Soviet Union Mathematical Olympiad, 483

Tags: broken line , parabola , combinatorial geometry , geometry , conic

A polygonal line with a finite number of segments has all its vertices on a parabola. Any two adjacent segments make equal angles with the tangent to the parabola at their point of intersection. One end of the polygonal line is also on the axis of the parabola. Show that the other vertices of the polygonal line are all on the same side of the axis.

1996 All-Russian Olympiad Regional Round, 8.6

Tags: geometry , combinatorics , combinatorial geometry , reflection

Spot spotlight located at vertex $B$ of an equilateral triangle $ABC$, illuminates angle $\alpha$. Find all such values of $\alpha$, not exceeding $60^o$, which at any position of the spotlight, when the illuminated corner is entirely located inside the angle $ABC$, from the illuminated and two unlit segments of side $AC$ can be formed into a triangle.

2020 Princeton University Math Competition, 14

Tags: combinatorics , combinatorial geometry , geometry

Let $N$ be the number of convex $27$-gons up to rotation there are such that each side has length $ 1$ and each angle is a multiple of $2\pi/81$. Find the remainder when $N$ is divided by $23$.

2006 All-Russian Olympiad Regional Round, 10.8

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

A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?

1980 All Soviet Union Mathematical Olympiad, 293

Tags: vector , plane , combinatorial geometry , geometry

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

2020 Costa Rica - Final Round, 6

Tags: combinatorics , combinatorial geometry , geometry

$10$ persons sit around a circular table and on the table there are $22$ vases. Two persons can see each other if and only if there are no vases aligned with them. Prove that there are at least two people who can see each other.

1979 Chisinau City MO, 181

Tags: collinear , geometry , combinatorial geometry , point

Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.

1996 IMO Shortlist, 4

Tags: combinatorial geometry , combinatorics , point set , triangle

Determine whether or nor there exist two disjoint infinite sets $ A$ and $ B$ of points in the plane satisfying the following conditions: a.) No three points in $ A \cup B$ are collinear, and the distance between any two points in $ A \cup B$ is at least 1. b.) There is a point of $ A$ in any triangle whose vertices are in $ B,$ and there is a point of $ B$ in any triangle whose vertices are in $ A.$

Russian TST 2018, P1

Tags: combinatorics , geometry , combinatorial geometry

Find all positive $r{}$ satisfying the following condition: For any $d > 0$, there exist two circles of radius $r{}$ in the plane that do not contain lattice points strictly inside them and such that the distance between their centers is $d{}$.

1947 Kurschak Competition, 3

Tags: combinatorics , disks , combinatorial geometry

What is the smallest number of disks radius $\frac12$ that can cover a disk radius $1$?

2016 Putnam, B3

Tags: putnam geometry , combinatorial geometry

Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$