Olympiad problems

Estonia Open Junior - geometry, 2014.2.5

In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?

1993 Bulgaria National Olympiad, 6

Tags: combinatorial geometry , combinatorics , points

Find all natural numbers $n$ for which there exists set $S$ consisting of $n$ points in the plane, satisfying the condition: For each point $A \in S$ there exist at least three points say $X, Y, Z$ from $S$ such that the segments $AX, AY$ and$ AZ$ have length $1$ (it means that $AX = AY = AZ = 1$).

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: combinatorics , lattice , points , Coloring

Let $M$ be the set of integer coordinate points situated on the line $d$ of real numbers. We color the elements of M in black or white. Show that at least one of the following statements is true: (a) there exists a finite subset $F \subset M$ and a point $M \in d$ so that the elements of the set $M - F$ that are lying on one of the rays determined by $M$ on $d$ are all white, and the elements of $M - F$ that are situated on the opposite ray are all black, (b) there exists an infinite subset $S \subset M$ and a point $T \in d$ so that for each $A \in S$ the reflection of A about $T$ belongs to $S$ and has the same color as $A$

1999 North Macedonia National Olympiad, 4

Tags: combinatorial geometry , lines , points

Do there exist $100$ straight lines on a plane such that they intersect each other in exactly $1999$ points?

1991 Denmark MO - Mohr Contest, 5

Tags: geometry , combinatorial geometry , combinatorics , circle , points

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

1978 Dutch Mathematical Olympiad, 3

Tags: points , combinatorics , combinatorial geometry

There are $1978$ points in the flat plane. Each point has a circular disk with that point as its center and the radius is the distance to a fixed point. Prove that there are five of these circular disks, which together cover all $1978$ points (circular disk means: the circle and its inner area).

2015 Ukraine Team Selection Test, 9

Tags: Heptagon , pentagon , convex , points , combinatorics , combinatorial geometry

The set $M$ consists of $n$ points on the plane and satisfies the conditions: $\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon, $\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon. Find the smallest possible value that $n$ can take .

2001 Junior Balkan Team Selection Tests - Moldova, 1

Tags: Coloring , points , combinatorial geometry , combinatorics

On a circle we consider a set $M$ consisting of $n$ ($n \ge 3$) points, of which only one is colored red. Determine of which polygons inscribed in a circle having the vertices in the set $M$ are more: those that contain the red dot or those that do not contain those points? How many more are there than others?

1968 Polish MO Finals, 5

Tags: geometry , points , combinatorics , combinatorial geometry

Given $n \ge 4$ points in the plane such that any four of them are the vertices of a convex quadrilateral, prove that these points are the vertices of a convex polygon.

2003 Portugal MO, 3

Tags: geometry , points , combinatorial geometry

Raquel painted $650$ points in a circle with a radius of $16$ cm. Shows that there is a circular crown with $2$ cm of inner radius and $3$ cm of outer radius that contain at least $10$ of these points.

1987 Polish MO Finals, 1

Tags: combinatorial geometry , combinatorics , geometry , square , points , distance

There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.

1949-56 Chisinau City MO, 58

Tags: geometry , combinatorial geometry , combinatorics , points , lines

On the plane $n$ points are chosen so that exactly $m$ of them lie on one straight line and no three points not included in these $m$ points lie on one straight line. What is the number of all lines, each of which contains at least two of these points?

1997 Poland - Second Round, 6

Tags: geometry , 3D geometry , combinatorial geometry , cube , distance , max , points

Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.

1994 North Macedonia National Olympiad, 4

Tags: combinatorics , points , circle

$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$. Determine the smallest number of circles with radius $1$, sufficient to cover all $1994$

Russian TST 2016, P1

Tags: combinatorics , Coloring , points , combinatorial geometry

$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$, and the sum of the lengths of the segments with the ends of different colors is $400$. Prove that you can draw a straight line separating everything red dots from all blue ones.