Olympiad problems

Novosibirsk Oral Geo Oly VII, 2020.3

Cut an arbitrary triangle into $2019$ pieces so that one of them turns out to be a triangle, one is a quadrilateral, ... one is a $2019$-gon and one is a $2020$-gon. Polygons do not have to be convex.

2023 Flanders Math Olympiad, 3

Tags: geometry , combinatorial geometry

The vertices of a regular $4$-gon, $6$-gon and $12$-goncan be brought together in one point to form a complete angle of $360^o$ (see figure). [center][img]https://cdn.artofproblemsolving.com/attachments/b/1/e9245179b7e0f5acb98b226bdc6db87fd72ad5.png[/img] [/center] Determine all triples $a, b, c \in N$ with $a < b < c$ for which the angles of a regular $a$-gon, $b$-gon and $c$-gon together also form $360^o$ .

2011 District Olympiad, 1

Tags: combinatorics , geometry , combinatorial geometry

In a square of side length $60$, $121$ distinct points are given. Show that among them there exists three points which are vertices of a triangle with an area not exceeding $30$.

2013 Abels Math Contest (Norwegian MO) Final, 4b

Tags: combinatorial geometry , combinatorics

A total of $a \cdot b \cdot c$ cubical boxes are joined together in a $a \times b \times c$ rectangular stack, where $a, b, c \ge 2$. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples $(a, b, c)$ is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?

2013 Romania National Olympiad, 2

Tags: combinatorics , combinatorial geometry , chessboard

A rook starts moving on an infinite chessboard, alternating horizontal and vertical moves. The length of the first move is one square, of the second – two squares, of the third – three squares and so on. a) Is it possible for the rook to arrive at its starting point after exactly $2013$ moves? b) Find all $n$ for which it possible for the rook to come back to its starting point after exactly $n$ moves.

1995 Tournament Of Towns, (453) 2

Tags: geometry , combinatorics , combinatorial geometry

Four grasshoppers sit at the vertices of a square. Every second, one of them jumps over one of the others to the symmetrical point on the other side (if $X$ jumps over $Y$ to the point $X'$, then $X$, $Y$ and $X'$ lie on a straight line and $XY = YX'$). Prove that after several jumps no three grasshoppers can be: (a) on a line parallel to a side of the square, (b) on a straight line. (AK Kovaldzhy)

Mathley 2014-15, 3

Tags: combinatorics , combinatorial geometry , isosceles

Given a regular $2013$-sided polygon, how many isosceles triangles are there whose vertices are vertices vertex of given polygon and haave an angle greater than $120^o$? Nguyen Tien Lam, High School for Natural Science,Hanoi National University.

2009 Romanian Master of Mathematics, 2

Tags: analytic geometry , modular arithmetic , combinatorics , number theory , combinatorial geometry

A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ [i]Dan Schwarz, Romania[/i]

1966 IMO Shortlist, 4

Tags: geometry , combinatorial geometry , combinatorics

Given $5$ points in the plane, no three of them being collinear. Show that among these $5$ points, we can always find $4$ points forming a convex quadrilateral.

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)

2016 Korea Summer Program Practice Test, 8

Tags: combinatorial geometry

There are distinct points $A_1, A_2, \dots, A_{2n}$ with no three collinear. Prove that one can relabel the points with the labels $B_1, \dots, B_{2n}$ so that for each $1 \le i < j \le n$ the segments $B_{2i-1} B_{2i}$ and $B_{2j-1} B_{2j}$ do not intersect and the following inequality holds. \[ B_1 B_2 + B_3 B_4 + \dots + B_{2n-1} B_{2n} \ge \frac{2}{\pi} (A_1 A_2 + A_3 A_4 + \dots + A_{2n-1} A_{2n}) \]

2019 Tournament Of Towns, 3

Tags: bicentric quadrilateral , tiling , combinatorics , combinatorial geometry , cut

Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed. (Nairi Sedrakyan)

1934 Eotvos Mathematical Competition, 3

Tags: combinatorics , combinatorial geometry

We are given an infinite set of rectangles in the plane, each with vertices of the form $(0, 0)$, $(0,m)$, $(n, 0)$ and $ (n,m)$, where $m$ and $n$ are positive integers. Prove that there exist two rectangles in the set such that one contains the other.

Mathley 2014-15, 1

Tags: combinatorics , combinatorial geometry

A large golden square land lot of dimension $100 \times 100$ m was subdivided into $100$ square lots, each measured $10\times10$ m. A king of landfill had his men dump wastes onto some of the lots. There was a practice that if a particular lot was not dumped and twoof its adjacents had waste materials, then the lot would be filled with wastes the next day by the people. One day if all the lotswere filled with wastes, the king would claim his ownership ofthe whole land lot. At least how many lots should have the kind had his men dump wastes onto? Vu Ha Van, Mathematics Faculty, Yale University, USA.

1987 Tournament Of Towns, (137) 2

Tags: octagon , tangential , combinatorial geometry , combinatorics

Quadrilaterals may be obtained from an octagon by cutting along its diagonals (in $8$ different ways) . Can it happen that among these $8$ quadrilaterals (a) four (b ) five possess an inscribed circle? (P. M . Sedrakyan , Yerevan)

1940 Moscow Mathematical Olympiad, 066

Tags: combinatorial geometry , geometry , cone , 3d geometry

* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.

2021 Baltic Way, 15

Tags: combinatorial geometry , geometry , combinatorics

For which positive integers $n\geq4$ does there exist a convex $n$-gon with side lengths $1, 2, \dots, n$ (in some order) and with all of its sides tangent to the same circle?

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?

1992 ITAMO, 1

Tags: combinatorial geometry , 3d geometry , cube , plane , geometry

A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.

2025 NCMO, 3

Tags: combinatorial geometry

Let $\mathcal{S}$ be a set of points in the plane such that for each subset $\mathcal{T}$ of $\mathcal{S}$, there exists a convex $2025$-gon which contains all of the points in $\mathcal{T}$ and none of the rest of the points in $\mathcal{S}$ but not $\mathcal{T}$. Determine the greatest possible number of points in $\mathcal{S}$. [i]Jason Lee[/i]

2011 Sharygin Geometry Olympiad, 8

Tags: combinatorial geometry , cube , geometry

Given a sheet of tin $6\times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?

2004 Iran MO (3rd Round), 26

Tags: 3d geometry , sphere , combinatorial geometry , geometry

Finitely many points are given on the surface of a sphere, such that every four of them lie on the surface of open hemisphere. Prove that all points lie on the surface of an open hemisphere.

1948 Moscow Mathematical Olympiad, 142

Tags: distance , combinatorics , combinatorial geometry

Find all possible arrangements of $4$ points on a plane, so that the distance between each pair of points is equal to either $a$ or $b$. For what ratios of $a : b$ are such arrangements possible?

1985 IMO Longlists, 41

Tags: combinatorics , point set , combinatorial geometry , ramsey theory

A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.

2021 Czech-Polish-Slovak Junior Match, 3

Tags: combinatorics , combinatorial geometry

A [i]cross [/i] is the figure composed of $6$ unit squares shown below (and any figure made of it by rotation). [img]https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png[/img] Find the greatest number of crosses that can be cut from a $6 \times 11$ divided sheet of paper into unit squares (in such a way that each cross consists of six such squares).