Olympiad problems

2014 Ukraine Team Selection Test, 10

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.

Taiwan TST 2015 Round 1, 3

Tags: convex hull , combinatorics

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

1999 IMO, 1

Tags: symmetry , 3d geometry , convex hull , reflection , geometry

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

2021 China Team Selection Test, 5

Tags: combinatorics , combinatorial geometry , area , convex hull , convex polygon , symmetry , geometry

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

2011 Miklós Schweitzer, 6

Tags: real analysis , euclidean space , convex hull , combinatorial geometry

Let $C_1, ..., C_d$ be compact and connected sets in $R^d$, and suppose that each convex hull of $C_i$ contains the origin. Prove that for every i there is a $c_i \in C_i$ for which the origin is contained in the convex hull of the points $c_1, ..., c_d$.

2002 IMO, 6

Tags: geometric inequality , geometry , convex hull , combinatorial geometry

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]

2015 Postal Coaching, Problem 2

Tags: combinatorics , convex hull

Given $2015$ points in the plane, show that if every four of them form a convex quadrilateral then the points are the vertices of a convex $2015-$sided polygon.

1999 IMO Shortlist, 3

Tags: reflection , symmetry , geometry , 3d geometry , convex hull

A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

2020 Silk Road, 4

Tags: combinatorial geometry , convex hull , combinatorics

Prove that for any natural number $ m $ there exists a natural number $ n $ such that any $ n $ distinct points on the plane can be partitioned into $ m $ non-empty sets whose [i]convex hulls[/i] have a common point. The [i] convex hull [/i] of a finite set $ X $ of points on the plane is the set of points lying inside or on the boundary of at least one convex polygon with vertices in $ X $, including degenerate ones, that is, the segment and the point are considered convex polygons. No three vertices of a convex polygon are collinear. The polygon contains its border.

2021 China Team Selection Test, 5

Tags: combinatorial geometry , symmetry , convex hull , convex polygon , area , combinatorics , geometry

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

Russian TST 2015, P4

Tags: combinatorics , convex hull

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

1987 Brazil National Olympiad, 4

Tags: minimum , combinatorial geometry , algebra , convex hull

Given points $A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), .., A_n (x_n, y_n, z_n)$ let $P (x, y, z)$ be the point which minimizes $\Sigma ( |x - x_i| + |y -y_i| + |z -z_i| )$. Give an example (for each $n > 4$) of points $A_i $ for which the point $P$ lies outside the convex hull of the points $A_i$.

2014 IMO Shortlist, C7

Tags: combinatorics , convex hull

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

2002 IMO Shortlist, 6

Tags: combinatorial geometry , geometry , geometric inequality , convex hull

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]