Olympiad problems

2016 Saudi Arabia IMO TST, 1

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 $d$.

1984 Tournament Of Towns, (072) 3

Tags: point , segment , combinatorial geometry , geometry

On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely? (V.E. Kolosov)

1986 China Team Selection Test, 4

Tags: point , chord , combinatorics

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.

2015 Ukraine Team Selection Test, 9

Tags: point , heptagon , pentagon , convex , 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 .

1953 Moscow Mathematical Olympiad, 251

Tags: point , convex polygon , combinatorial geometry , circles

On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?

2008 Thailand Mathematical Olympiad, 10

Tags: point , combinatorics , combinatorial geometry

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.

1968 Polish MO Finals, 5

Tags: point , geometry , 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.

2008 Postal Coaching, 6

Tags: point , combinatorics , combinatorial geometry , equilateral

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

1993 Bulgaria National Olympiad, 6

Tags: point , combinatorial geometry , combinatorics

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 $AX = AY = AZ = 1$).

2010 Contests, 1

Tags: point , combinatorics , combinatorial geometry , 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.

1998 All-Russian Olympiad Regional Round, 10.3

Tags: point , combinatorics , combinatorial geometry , geometry

Prove that from any finite set of points on the plane, you can remove a point from the bottom in such a way that the remaining set can be split into two parts of smaller diameter. (Diameter is the maximum distance between points of the set.) [hide=original wording]Докажите, что из любого конечного множества точек на плоскости можно так удалитьо дну точку, что оставшееся множество можно разбить на две части меньшего диаметра. (Диаметр—это максимальное расстояние между точками множества.)[/hide]

1969 Polish MO Finals, 6

Tags: point , geometry , combinatorial geometry

Given a set $n$ of points in the plane that are not contained in a single straight line. Prove that there exists a circle passing through at least three of these points, inside which there are none of the remaining points of the set.

1982 Tournament Of Towns, (026) 4

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

(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, 1

Tags: point , combinatorial geometry , combinatorics , 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?

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.

1962 Putnam, A1

Tags: point , geometry , convex

Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.

2012 Tournament of Towns, 2

Tags: point , combinatorics , combinatorial geometry , circles

One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.

2001 Tuymaada Olympiad, 1

Tags: point , combinatorics , 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?

2015 Peru MO (ONEM), 1

Tags: point , geometry , combinatorics , combinatorial geometry

If $C$ is a set of $n$ points in the plane that has the following property: For each point $P$ of $C$, there are four points of $C$, each one distinct from $P$ , which are the vertices of a square. Find the smallest possible value of $n$.

Russian TST 2016, P1

Tags: point , combinatorics , coloring , 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.

1981 Poland - Second Round, 5

Tags: point , combinatorics , geometry , combinatorial geometry

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 $.

2014 Chile National Olympiad, 3

Tags: point , combinatorics , combinatorial geometry

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.

1978 Chisinau City MO, 163

Tags: point , polyline , geometry , combinatorial geometry , geometric inequality

On the plane $n$ points are selected that do not belong to one straight line. Prove that the shortest closed path passing through all these points is a non-self-intersecting polygon.

2002 Switzerland Team Selection Test, 1

Tags: point , plane , combinatorics , combinatorial geometry

In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.

1977 Czech and Slovak Olympiad III A, 5

Tags: combinatorial geometry , coloring , point

Let $A_1,\ldots,A_n$ be different collinear points. Every point is dyed by one of four colors and every of these colors is used at least once. Show that there is a line segment where two colors are used exactly once and the other two are used at least once.