This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 1704

2024 Moldova Team Selection Test, 5
Tags: combinatorial geometry
Consider a natural number $n \ge 3$. A convex polygon with $n$ sides is entirely placed inside a square with side length 1. Prove that we can always find three vertices of this polygon, the triangle formed by which has area not greater than $\frac{8}{n^2}$.

Novosibirsk Oral Geo Oly VII, 2022.7
Tags: geometry , combinatorics , combinatorial geometry
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?

2021 Hong Kong TST, 1
Tags: combinatorics , combinatorial geometry
Let $S$ be a set of $2020$ distinct points in the plane. Let \[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\] Find the least possible value of the number of points in $M$.

2021 Sharygin Geometry Olympiad, 8.8
Tags: convex polygon , combinatorial geometry , obtuse triangle , equal sides , geometry
Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

2014 Ukraine Team Selection Test, 10
Tags: point , convex hull , combinatorics , combinatorial geometry
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.

2021 Regional Olympiad of Mexico West, 6
Tags: rectangle , tiles , geometry , combinatorial geometry , combinatorics
Let $n$ be an integer greater than $3$. Show that it is possible to divide a square into $n^2 + 1$ or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of $n$ rectangles. Note: We say that two rectangles are [i]disjointed [/i] if they do not intersect or only intersect at their perimeters.

2004 Iran MO (3rd Round), 4
Tags: combinatorics , combinatorial geometry
We have finite white and finite black points that for each 4 oints there is a line that white points and black points are at different sides of this line.Prove there is a line that all white points and black points are at different side of this line.

1978 Austrian-Polish Competition, 6
Tags: circles , combinatorics , combinatorial geometry
We are given a family of discs in the plane, with pairwise disjoint interiors. Each disc is tangent to at least six other discs of the family. Show that the family is infinite.

1984 IMO Longlists, 47
Tags: coloring , ramsey theory , combinatorial geometry , geometry
Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

1975 Putnam, B2
Tags: combinatorial geometry , 3d geometry
A [i]slab[/i] is the set of points strictly between two parallel planes. Prove that a countable sequence of slabs, the sum of whose thicknesses converges, cannot fill space.

2008 Pre-Preparation Course Examination, 2
Tags: probability and stats , probability , combinatorial geometry
Seven points are selected randomly from $ S^1\subset\mathbb C$. What is the probability that origin is not contained in convex hull of these points?

1997 Poland - Second Round, 3
Tags: combinatorial geometry , segment , coloring , combinatorics
Let be given $n$ points, no three of which are on a line. All the segments with endpoints in these points are colored so that two segments with a common endpoint are of different colors. Determine the least number of colors for which this is possible

2022 Swedish Mathematical Competition, 1
Tags: combinatorics , tiling , tiles , combinatorial geometry
What sizes of squares with integer sides can be completely covered without overlap by identical tiles consisting of three squares with side $1$ joined together in one $L$ shape? [center][img]https://cdn.artofproblemsolving.com/attachments/3/f/9fe95b05527857f7e44dfd033e6fb01e5d25a2.png[/img][/center]

2006 Sharygin Geometry Olympiad, 8.2
Tags: polygon , cut , geometry , combinatorial geometry , minimum
What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?

2016 Saint Petersburg Mathematical Olympiad, 5
Tags: coloring , winning positions , game , game strategy , combinatorial geometry , combinatorics
Kostya and Sergey play a game on a white strip of length 2016 cells. Kostya (he plays first) in one move should paint black over two neighboring white cells. Sergey should paint either one white cell either three neighboring white cells. It is forbidden to make a move, after which a white cell is formed the doesn't having any white neighbors. Loses the one that can make no other move. However, if all cells are painted, then Kostya wins. Who will win if he plays the right game (has a winning strategy)?

1989 Tournament Of Towns, (231) 5
Tags: tiling , combinatorial geometry , combinatorics
A rectangular $M \times N$ board is divided into $1 \times $ cells. There are also many domino pieces of size $1 \times 2$. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is (a) less than $\frac14 MN$, (b) less than $\frac15 MN$.

2019 Hong Kong TST, 2
Tags: probability , combinatorial geometry , combinatorics
A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}.$ Find the sum of the largest and smallest possible value of $x$.

1947 Moscow Mathematical Olympiad, 138
Tags: points in space , combinatorial geometry , 3d geometry , geometry
In space, $n$ wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of $k$ triangles. Find all $n$ and $k$ for which this is possible.

2019 China Team Selection Test, 3
Tags: geometry , combinatorics , combinatorial geometry
$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

1983 Bundeswettbewerb Mathematik, 4
Tags: combinatorics , distance , combinatorial geometry
Let $g$ be a straight line and $n$ a given positive integer. Prove that there are always n different points on g to choose as well as a point not lying on g in such a way that the distance between each two of these $n + 1$ points is an integer.

1966 Czech and Slovak Olympiad III A, 2
Tags: geometry , combinatorial geometry , circles
Into how many regions do $n$ circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?

2012 IMAR Test, 1
Tags: vector , combinatorial geometry , set , vector geometry , geometry
Let $K$ be a convex planar set, symmetric about a point $O$, and let $X, Y , Z$ be three points in $K$. Show that $K$ contains the head of one of the vectors $\overrightarrow{OX} \pm \overrightarrow{OY} , \overrightarrow{OX} \pm \overrightarrow{OZ}, \overrightarrow{OY} \pm \overrightarrow{OZ}$.

2023 Olympic Revenge, 4
Tags: 3d geometry , lattice points , geometry , sphere , combinatorics , combinatorial geometry
Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?

2024 Romanian Master of Mathematics, 3
Tags: linear algebra , set , geometry , combinatorial geometry , combinatorics
Given a positive integer $n$, a collection $\mathcal{S}$ of $n-2$ unordered triples of integers in $\{1,2,\ldots,n\}$ is [i]$n$-admissible[/i] if for each $1 \leq k \leq n - 2$ and each choice of $k$ distinct $A_1, A_2, \ldots, A_k \in \mathcal{S}$ we have $$ \left|A_1 \cup A_2 \cup \cdots A_k \right| \geq k+2.$$ Is it true that for all $n > 3$ and for each $n$-admissible collection $\mathcal{S}$, there exist pairwise distinct points $P_1, \ldots , P_n$ in the plane such that the angles of the triangle $P_iP_jP_k$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$? [i]Ivan Frolov, Russia[/i]

1976 All Soviet Union Mathematical Olympiad, 220
Tags: distance , combinatorics , combinatorial geometry
There are $50$ exact watches lying on a table. Prove that there exist a certain moment, when the sum of the distances from the centre of the table to the ends of the minute hands is more than the sum of the distances from the centre of the table to the centres of the watches.