Olympiad problems

1967 All Soviet Union Mathematical Olympiad, 087

a) Can you pose the numbers $0,1,...,9$ on the circumference in such a way, that the difference between every two neighbours would be either $3$ or $4$ or $5$? b) The same question, but about the numbers $0,1,...,13$.

1972 Poland - Second Round, 4

Tags: combinatorial geometry , 3d geometry , geometry , combinatorics

A cube with edge length $ n $ is divided into $ n^3 $ unit cubes by planes parallel to its faces. How many pairs of such unit cubes exist that have no more than two vertices in common?

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 $\frac{8}{n^2}$.

2022 Novosibirsk Oral Olympiad in Geometry, 5

Tags: geometry , combinatorial geometry

Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.

Russian TST 2020, P3

Tags: combinatorics , combinatorial geometry , angle

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

2011 QEDMO 8th, 6

Tags: combinatorial geometry , geometry , combinatorics

A [i]synogon [/i] is a convex $2n$-gon with all sides of the same length and all opposite sides are parallel. Show that every synogon can be broken down into a finite number of rhombuses.

2021 Caucasus Mathematical Olympiad, 7

Tags: combinatorial geometry , combinatorics

4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to get three collinear tokens. Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves tokens were at the vertices of one another rectangle $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not equal to $\Pi $. Prove that $\Pi$ is a square.

1961 Kurschak Competition, 1

Tags: point , ratio , combinatorics , combinatorial geometry

Given any four distinct points in the plane, show that the ratio of the largest to the smallest distance between two of them is at least $\sqrt2$.

1986 Tournament Of Towns, (128) 3

Tags: combinatorial geometry , combinatorics , cover , triangle

Does there exist a set of $100$ triangles in which not one of the triangles can be covered by the other $99$?

1976 All Soviet Union Mathematical Olympiad, 230

Tags: equilateral , combinatorial geometry , combinatorics

Let us call "[i]big[/i]" a triangle with all sides longer than $1$. Given a equilateral triangle with all the sides equal to $5$. Prove that: a) You can cut $100$ [i]big [/i] triangles out of given one. b) You can divide the given triangle onto $100$ [i]big [/i] nonintersecting ones fully covering the initial one. c) The same as b), but the triangles either do not have common points, or have one common side, or one common vertex. d) The same as c), but the initial triangle has the side $3$.

2014 Junior Balkan Team Selection Tests - Romania, 4

Tags: equilateral , combinatorial geometry , combinatorics

On each side of an equilateral triangle of side $n \ge 1$ consider $n - 1$ points that divide the sides into $n$ equal segments. Through these points draw parallel lines to the sides of the triangles, obtaining a net of equilateral triangles of side length $1$. On each of the vertices of the small triangles put a coin head up. A move consists in flipping over three mutually adjacent coins. Find all values of $n$ for which it is possible to turn all coins tail up after a finite number of moves. Colombia 1997

2000 All-Russian Olympiad Regional Round, 8.3

Tags: combinatorics , combinatorial geometry , geometry

What is the smallest number of sides that an polygon can have (not necessarily convex), which can be cut into parallelograms?

1939 Moscow Mathematical Olympiad, 053

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

What is the greatest number of parts that $5$ spheres can divide the space into?

2021 239 Open Mathematical Olympiad, 7

Tags: geometry , combinatorial geometry

Given $n$ lines on the plane, they divide the plane onto several bounded or bounded polygonal regions. Define the rank of a region as the number of vertices on its boundary (a vertex is a point which belongs to at least two lines). Prove that the sum of squares of ranks of all regions does not exceed $10n^2$. (D. Fomin)

Kvant 2020, M2620

Tags: geometry , combinatorial geometry

A satellite is considered accessible from the point $A{}$ of the planet's surface if it is located relative to the tangent plane drawn at point $A{}$, strictly on the other side than the planet. What is the smallest number of satellites that need to be launched over a spherical planet so that at some point the signals of at least two satellites are available from each point on the planet's surface? [i]Proposed by S. Volchenkov[/i]

I Soros Olympiad 1994-95 (Rus + Ukr), 10.9

Tags: combinatorics , geometry , combinatorial geometry , cyclic quadrilateral

Prove that for all natural $n\ge 6 000$ any convex $1994$-gon can be cut into $n$ such quadrilaterals thata circle can be circumscribed around each of them

2017 ELMO Shortlist, 3

Tags: geometry , combinatorial geometry

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs. [i]Proposed by Daniel Liu

MIPT student olimpiad autumn 2024, 4

Tags: analytic geometry , geometric inequality , combinatorial geometry

The ellipsoid $E$ is contained in the simplex $S$, which is located in the unit ball B space $R^n$. Prove that the sum of the principal semi-axes of the ellipsoid $E$ is no more than units.

1978 All Soviet Union Mathematical Olympiad, 263

Tags: combinatorial geometry

Given $n$ nonintersecting segments in the plane. Not a pair of those belong to the same straight line. We want to add several segments, connecting the ends of given ones, to obtain one nonselfintersecting broken line. Is it always possible?

2017 ELMO Shortlist, 3

Tags: geometry , combinatorial geometry

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs. [i]Proposed by Daniel Liu

2020 Estonia Team Selection Test, 2

Tags: combinatorics , combinatorial geometry , min , angle , geometry

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

Novosibirsk Oral Geo Oly VII, 2022.7

Tags: combinatorial geometry , combinatorics , 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?

I Soros Olympiad 1994-95 (Rus + Ukr), 9.6

Tags: combinatorial geometry , circles , hexagon , geometry

Given a regular hexagon, whose sidelength is $ 1$ . What is the largest number of circles of radius $\frac{\sqrt3}{4}$ can be placed without overlapping inside such a hexagon? (Circles can touch each other and the sides of the hexagon.)

2022 Putnam, B2

Tags: linear algebra , combinatorial geometry

Let $\times$ represent the cross product in $\mathbb{R}^3.$ For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $$S=\{v \times w: v, w \in S\}?$$

2008 IMO Shortlist, 5

Tags: geometry , point set , combinatorial geometry , line

Let $ k$ and $ n$ be integers with $ 0\le k\le n \minus{} 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k \plus{} 1)(k \plus{} 2)$ red points. [i]Proposed by Gerhard Woeginger, Netherlands[/i]