Olympiad problems

VI Soros Olympiad 1999 - 2000 (Russia), 10.2

$37$ points are arbitrarily marked on the plane. Prove that among them there must be either two points at a distance greater than $6$, or two points at a distance less than $1.5$.

1997 Slovenia Team Selection Test, 3

Tags: combinatorics , combinatorial geometry , coloring , point

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

2010 Sharygin Geometry Olympiad, 8

Tags: geometry , combinatorics , polygon , regular polygon , perimeter , minimum , point

Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.

2016 Singapore Junior Math Olympiad, 5

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

Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points. (Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$)

2009 IMAC Arhimede, 6

Tags: combinatorics , point

At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw score and zero points for the defeat. At the end of the tournament it turned out that the sum of the winning points of all teams was $50$. (a) How many teams participated in this tournament? (b) How big is the difference between the team with the highest number and the number of points won?

2018 Estonia Team Selection Test, 11

Tags: combinatorics , combinatorial geometry , point

Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied: 1) there is at least $1$ marked point on each side, 2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$

2014 Contests, 4

Tags: number theory , point , circles , prime

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

1994 Tuymaada Olympiad, 1

Tags: point , combinatorics

World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$

1979 All Soviet Union Mathematical Olympiad, 283

Tags: combinatorics , collinear , point

Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2$, $A_2A_3$,$ ...$, $A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$. Prove that it is possible a) for $k=3$, b) for every $k$ less than $(n-1)$.

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.

1999 Korea Junior Math Olympiad, 7

Tags: geometry , combinatorics , point

$A_0B, A_0C$ rays that satisfy $\angle BA_0C=14^{\circ}$. You are to place points $A_1, A_2, ...$ by the following rules. [b]Rules[/b] (1) On the first move, place $A_1$ on any point on $A_0B$(except $A_0$). (2) On the $n>1$th move, place $A_n$ on $A_0B$ iff $A_{n-1}$ is on $A_0C$, and place $A_n$ on $A_0C$ iff $A_{n-1}$ is one $A_0B$. $A_n$ must be place on the point that satisfies $A_{n-2}A_n{n-1}=A_{n-1}A_n$. All the points must be placed in different locations. What is the maximum number of points that can be placed?

Russian TST 2016, P1

Tags: combinatorics , coloring , combinatorial geometry , 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.

1986 China Team Selection Test, 4

Tags: combinatorics , point , chord

Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

1982 Tournament Of Towns, (026) 4

Tags: geometry , combinatorial geometry , circles , arc , chord , point

(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords. Is it necessary that two of these chords are of equal length? (b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords. Prove that among these $10$ chords there are two chords of equal length. (VV Proizvolov, Moscow)

2011 Tournament of Towns, 7

Tags: combinatorial geometry , combinatorics , circles , point

$100$ 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.

1950 Moscow Mathematical Olympiad, 184

Tags: combinatorics , combinatorial geometry , point

* On a circle, $20$ points are chosen. Ten non-intersecting chords without mutual endpoints connect some of the points chosen. How many distinct such arrangements are there?

2015 JBMO Shortlist, C2

Tags: combinatorics , combinatorial geometry , distance , point

$2015$ 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.

2014 NZMOC Camp Selection Problems, 4

Tags: combinatorial geometry , combinatorics , point

Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?

1981 Brazil National Olympiad, 4

Tags: combinatorial geometry , graph , combinatorics , point

A graph has $100$ points. Given any four points, there is one joined to the other three. Show that one point must be joined to all $99$ other points. What is the smallest number possible of such points (that are joined to all the others)?

1986 China Team Selection Test, 4

Tags: combinatorics , point , chord

Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

2008 Postal Coaching, 6

Tags: combinatorics , combinatorial geometry , equilateral , point

A set of points in the plane is called [i]free [/i] if no three points of the set are the vertices of an equilateral triangle. Prove that any set of $n$ points in the plane has a free subset of at least $\sqrt{n}$ points

2011 IFYM, Sozopol, 4

Tags: plane , line , point , geometry

There are $n$ points in a plane. Prove that there exist a point $O$ (not necessarily from the given $n$) such that on each side of an arbitrary line, through $O$, lie at least $\frac{n}{3}$ points (including the points on the line).

Kyiv City MO Juniors Round2 2010+ geometry, 2019.7.31

Tags: geometry , construction , analytic geometry , coordinate geometry , point

The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$. Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$? Point A on the board located above and to the left of point $B$.

1990 All Soviet Union Mathematical Olympiad, 513

Tags: edge , graph , combinatorics , combinatorial geometry , point

A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.

2011 German National Olympiad, 4

Tags: angle , maximal , sum , set , point , geometry

There are two points $A$ and $B$ in the plane. a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.