Olympiad problems

1997 Poland - Second Round, 6

Tags: combinatorial geometry , geometry , 3d geometry , cube , distance , max , point

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

2010 IMAC Arhimede, 1

Tags: combinatorial geometry , combinatorics , point , collinear

$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.

1993 Bulgaria National Olympiad, 6

Tags: combinatorial geometry , combinatorics , point

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 $AX = AY = AZ = 1$).

2014 Chile National Olympiad, 3

Tags: combinatorial geometry , combinatorics , point

In the plane there are $2014$ plotted points, such that no $3$ are collinear. For each pair of plotted points, draw the line that passes through them. prove that for every three of marked points there are always two that are separated by an amount odd number of lines.

2002 Junior Balkan Team Selection Tests - Moldova, 2

Tags: combinatorics , combinatorial geometry , point , line

$64$ distinct points are positioned in the plane so that they determine exactly $2003$ different lines. Prove that among the $64$ points there are at least $4$ collinear points.

2001 Tuymaada Olympiad, 1

Tags: combinatorics , point , game

$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

2011 Tournament of Towns, 7

Tags: circles , combinatorial geometry , point , combinatorics

$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.

Denmark (Mohr) - geometry, 1991.5

Tags: geometry , combinatorial geometry , combinatorics , circles , point

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.

1969 Swedish Mathematical Competition, 6

Tags: combinatorics , combinatorial geometry , point , triangle

Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?

1949-56 Chisinau City MO, 58

Tags: combinatorics , combinatorial geometry , point , line , geometry

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?

2016 Latvia Baltic Way TST, 12

Tags: point , geometry

For what positive numbers $m$ and $n$ do there exist points $A_1, ..., Am$ and $B_1 ..., B_n$ in the plane such that, for any point $P$, the equation $$|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2 $$ holds true?

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.

2009 Moldova National Olympiad, 7.3

Tags: point , geometry

On the lines $AB$ are located $2009$ different points that do not belong to the segment $[AB]$. Prove that the sum of the distances from point $A$ to these points is not equal to the sum of the distances from point $B$ to these points.

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 $\frac12$.

Russian TST 2016, P1

Tags: combinatorics , combinatorial geometry , coloring , point

$101$ 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.

1997 Slovenia Team Selection Test, 3

Tags: combinatorics , combinatorial geometry , point , coloring

Let $A_1,A_2,...,A_n$ be $n \ge 2$ distinct points on a circle. Find the number of colorings of these points with $p \ge 2$ colors such that every two adjacent points receive different colors

1999 Singapore MO Open, 1

Tags: combinatorics , even , coloring , point

Let $n$ be a positive integer. A square $ABCD$ is divided into $n^2$ identical small squares by drawing $(n-1)$ equally spaced lines parallel to the side $AB$ and another $(n- 1)$ equally spaced lines parallel to $BC$, thus giving rise to $(n+1)^2$ intersection points. The points $A, C$ are coloured red and the points $B, D$ are coloured blue. The rest of the intersection points are coloured either red or blue. Prove that the number of small squares having exactly $3$ vertices of the same colour is even.

1961 Kurschak Competition, 1

Tags: combinatorial geometry , combinatorics , point , ratio

Given any four distinct points in the plane, show that the ratio of the largest to the smallest distance between two of them is at least $\sqrt2$.

1987 Polish MO Finals, 1

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

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 $P_n$.

1981 Poland - Second Round, 5

Tags: geometry , combinatorics , combinatorial geometry , point

In the plane there are two disjoint sets $ A $ and $ B $, each of which consists of $ n $ points, and no three points of the set $ A \cup B $ lie on one straight line. Prove that there is a set of $ n $ disjoint closed segments, each of which has one end in the set $ A $ and the other in the set $ B $.

2015 JBMO Shortlist, C2

Tags: combinatorics , point , combinatorial geometry , distance

$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.