Olympiad problems

2018 Iran MO (1st Round), 4

Tags: points

There are $5$ points in the plane no three of which are collinear. We draw all the segments whose vertices are these points. What is the minimum number of new points made by the intersection of the drawn segments? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$

1956 Moscow Mathematical Olympiad, 324

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

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

2014 Ukraine Team Selection Test, 10

Tags: combinatorial geometry , combinatorics , points , Convex hull

Find all positive integers $n \ge 4$ for which there are $n$ points in general position on the plane such that an arbitrary triangle with vertices belonging to the convex hull of these $n$ points, containing exactly one of $n - 3$ points inside remained.

2011 IFYM, Sozopol, 4

Tags: geometry , Plane , points , lines

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).

2011 German National Olympiad, 4

Tags: geometry , angle , Maximal , Sets , points , Sum

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.

2008 Postal Coaching, 6

Tags: combinatorics , combinatorial geometry , Equilateral , points

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

1999 Korea Junior Math Olympiad, 7

Tags: points , geometry , combinatorics , KJMO

$A_0B, A_0C$ 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?

2018 Estonia Team Selection Test, 11

Tags: combinatorics , combinatorial geometry , points

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 $Y$

1999 Singapore MO Open, 1

Tags: Even , points , Coloring , combinatorics

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.

1986 Tournament Of Towns, (131) 7

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

On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

2012 Tournament of Towns, 2

Tags: points , combinatorial geometry , combinatorics

One hundred points are marked in the plane, with no three in a line. Is it always possible to connect the points in pairs such that all fifty segments intersect one another?

2013 Tournament of Towns, 1

Tags: combinatorics , combinatorial geometry , points

There are six points on the plane such that one can split them into two triples each creating a triangle. Is it always possible to split these points into two triples creating two triangles with no common point (neither inside, nor on the boundary)?

1979 Bundeswettbewerb Mathematik, 3

Tags: combinatorics , points

The $n$ participants of a tournament are numbered with $0$ through $n - 1$. At the end of the tournament it turned out that for every team, numbered with $s$ and having $t$ points, there are exactly $t$ teams having $s$ points each. Determine all possibilities for the final score list.

2015 JBMO Shortlist, C2

Tags: combinatorics , combinatorial geometry , points , 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.

2022 Sharygin Geometry Olympiad, 8.7

Tags: geometry , points , combinatorics , combinatorial geometry

Ten points on a plane a such that any four of them lie on the boundary of some square. Is obligatory true that all ten points lie on the boundary of some square?

2011 Tournament of Towns, 1

Tags: combinatorial geometry , combinatorics , points , distance

Pete has marked several (three or more) points in the plane such that all distances between them are different. A pair of marked points $A,B$ will be called unusual if $A$ is the furthest marked point from $B$, and $B$ is the nearest marked point to $A$ (apart from $A$ itself). What is the largest possible number of unusual pairs that Pete can obtain?

2015 Finnish National High School Mathematics Comp, 5

Tags: points , correct , probability , combinatorics

Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?

1979 Chisinau City MO, 181

Tags: collinear , geometry , points , combinatorial geometry

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.

2016 Saint Petersburg Mathematical Olympiad, 4

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

$N> 4$ points move around the circle, each with a constant speed. For Any four of them have a moment in time when they all meet. Prove that is the moment when all the points meet.

1973 Chisinau City MO, 63

Tags: combinatorics , Coloring , points

Each point in space is colored in one of four different colors. Prove that there is a segment $1$ cm long with endpoints of the same color.

2010 IMAC Arhimede, 5

Tags: combinatorics , combinatorial geometry , points , Obtuse triangle

Different points $A_1, A_2,..., A_n$ in the plane ($n> 3$) are such that the triangle $A_iA_jA_k$ is obtuse for all the different $i,j,k \in\{1,2,...,n\}$. Prove that there is a point $A_{n + 1}$ in the plane, such that the triangle $A_iA_jA_{n + 1}$ is obtuse for all different $i,j \in\{1,2,...,n\}$

1997 Slovenia Team Selection Test, 3

Tags: combinatorics , combinatorial geometry , points , 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

2016 Saudi Arabia IMO TST, 1

Tags: combinatorial geometry , recurrence relation , points

On the Cartesian coordinate system $Oxy$, consider a sequence of points $A_n(x_n, y_n)$ in which $(x_n)^{\infty}_{n=1}$,$(y_n)^{\infty}_{n=1}$ are two sequences of positive numbers satisfing the following conditions: $$x_{n+1} =\sqrt{\frac{x_n^2+x_{n+2}^2}{2}}, y_{n+1} =\big( \frac{\sqrt{y_n}+\sqrt{y_{n+2}}}{2} \big)^2 \,\, \forall n \ge 1 $$ Suppose that $O, A_1, A_{2016}$ belong to a line $d$ and $A_1, A_{2016}$ are distinct. Prove that all the points $A_2, A_3,. .. , A_{2015}$ lie on one side of $d$.

1998 Estonia National Olympiad, 5

Tags: circles , points , Coloring , combinatorics , combinatorial geometry

A circle is divided into $n$ equal arcs by $n$ points. Assume that, no matter how we color the $n$ points in two colors, there always exists an axis of symmetry of the set of points such that any two of the $n$ points which are symmetric with respect to that axis have the same color. Find all possible values of $n$.

2003 Junior Balkan Team Selection Tests - Romania, 4

Tags: Coloring , combinatorics , combinatorial geometry , points

Show that one can color all the points of a plane using only two colors such that no line segment has all points of the same color.