Olympiad problems

1987 All Soviet Union Mathematical Olympiad, 461

All the faces of a convex polyhedron are the triangles. Prove that it is possible to paint all its edges in red and blue colour in such a way, that it is possible to move from the arbitrary vertex to every vertex along the blue edges only and along the red edges only.

2013 Saudi Arabia GMO TST, 2

Tags: polygon , geometry , combinatorial geometry , angle , equal angles , combinatorics

Find all values of $n$ for which there exists a convex cyclic non-regular polygon with $n$ vertices such that the measures of all its internal angles are equal.

1970 Kurschak Competition, 3

Tags: coloring , combinatorics , combinatorial geometry

n points are taken in the plane, no three collinear. Some of the line segments between the points are painted red and some are painted blue, so that between any two points there is a unique path along colored edges. Show that the uncolored edges can be painted (each edge either red or blue) so that all triangles have an odd number of red sides.

2012 Czech-Polish-Slovak Junior Match, 4

Tags: polygon , combinatorial geometry , regular polygon , isosceles , combinatorics

Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.

1980 All Soviet Union Mathematical Olympiad, 293

Tags: geometry , combinatorial geometry , plane , vector

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

2009 Germany Team Selection Test, 2

Tags: point set , line , combinatorial geometry , geometry

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]

1982 Czech and Slovak Olympiad III A, 3

Tags: combinatorial geometry , lattice points , geometry , combinatorics

In the plane with coordinates $x,y$, find an example of a convex set $M$ that contains infinitely many lattice points (i.e. points with integer coordinates), but at the same time only finitely many lattice points from $M$ lie on each line in that plane.

2025 Sharygin Geometry Olympiad, 13

Tags: combinatorial geometry , geometry

Each two opposite sides of a convex $2n$-gon are parallel. (Two sides are opposite if one passes $n-1$ other sides moving from one side to another along the borderline of the $2n$-gon.) The pair of opposite sides is called regular if there exists a common perpendicular to them such that its endpoints lie on the sides and not on their extensions. Which is the minimal possible number of regular pairs? Proposed by: B.Frenkin

1966 IMO Longlists, 49

Tags: geometry , combinatorial geometry , reflection

Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

2016 IFYM, Sozopol, 3

Tags: combinatorial geometry , combinatorics

Let $A_1 A_2…A_{66}$ be a convex 66-gon. What’s the greatest number of pentagons $A_i A_{i+1} A_{i+2} A_{i+3} A_{i+4},1\leq i\leq 66,$ which have an inscribed circle? ($A_{66+i}\equiv A_i$).

1995 Denmark MO - Mohr Contest, 5

Tags: combinatorial geometry , geometry , circles

In the plane, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.

1970 IMO Longlists, 26

Tags: combinatorial geometry , linear algebra , vector

Consider a finite set of vectors in space $\{a_1, a_2, ... , a_n\}$ and the set $E$ of all vectors of the form $x=\sum_{i=1}^{n}{\lambda _i a_i}$, where $\lambda _i \in \mathbb{R}^{+}\cup \{0\}$. Let $F$ be the set consisting of all the vectors in $E$ and vectors parallel to a given plane $P$. Prove that there exists a set of vectors $\{b_1, b_2, ... , b_p\}$ such that $F$ is the set of all vectors $y$ of the form $y=\sum_{i=1}^{p}{\mu _i b_i}$, where $\mu _i \in \mathbb{R}^{+}\cup \{0\}$.

1997 German National Olympiad, 5

Tags: combinatorics , geometry , area , combinatorial geometry

We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$. Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$.

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]

2006 Hong Kong TST., 5

Tags: combinatorial geometry , geometry

Given finitely many points in a plane, it is known that the area of the triangle formed by any three points of the set is less than 1. Show that all points of the set lie inside or on boundary of a triangle with area less than 4.

2015 Estonia Team Selection Test, 8

Tags: combinatorics , combinatorial geometry , regular polygon , diagonal

Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.

2022 Durer Math Competition (First Round), 2

Tags: combinatorial geometry , geometry , congruent triangles

Determine all triangles that can be split into two congruent pieces by one cut. A cut consists of segments $P_1P_2$, $P_2P_3$, . . . , $P_{n-1}P_n$ where points $P_1, P_2, . . . , P_n$ are distinct, points $P_1$ and $P_n$ lie on the perimeter of the triangle and the rest of the points lie in the interior of the triangle such that the segments are disjoint except for the endpoints.

1998 Tournament Of Towns, 4

Tags: combinatorial geometry , diagonal , combinatorics , concurrency

All the diagonals of a regular $25$-gon are drawn. Prove that no $9$ of the diagonals pass through one interior point of the $25$-gon. (A Shapovalov)

2021 Iran MO (2nd Round), 4

Tags: combinatorics , combinatorial geometry , geometry

$n$ points are given on a circle $\omega$. There is a circle with radius smaller than $\omega$ such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of $\omega$ with endpoints not belonging to the given points such that all the $n$ given points remain in one side of the diameter.

2025 Czech-Polish-Slovak Junior Match., 2

Tags: geometry , combinatorial geometry

Find all triangles that can be divided into congruent right-angled isosceles triangles with side lengths $1, 1, \sqrt{2}$.

2015 Estonia Team Selection Test, 2

Tags: rectangle , combinatorial geometry , combinatorics , square

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

1992 Tournament Of Towns, (322) 3

Tags: combinatorial geometry , combinatorics

A numismatist Fred has some coins. A diameter of any coin is no more than $10$ cm. All the coins are contained in a one-layer box of dimensions $30$ cm by $70$ cm. He is presented with a new coin. Its diameter is $25$ cm. Prove that it is possible to put all the coins in a one-layer box of dimensions $55$ cm by $55$ cm. (Fedja Nazarov, St Petersburg)

2011 QEDMO 8th, 8

Tags: combinatorial geometry , lattice , coloring , combinatorics

Albatross and Frankinfueter are playing again: each of them takes turns choosing one point in the plane with integer coordinates and paint it in his favorite color. Albatross plays first. Someone wins as soon as there is a square with all four corners in the are colored in their own color. Does anyone has a winning strategy and if so, who?

2011 Tournament of Towns, 5

Tags: combinatorial geometry , angle , sum , line , combinatorics

In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?

2017 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: combinatorics , combinatorial geometry , geometry

One have $n$ distinct circles(with the same radius) such that for any $k+1$ circles there are (at least) two circles that intersects in two points. Show that for each line $l$ one can make $k$ lines, each one parallel with $l$, such that each circle has (at least) one point of intersection with some of these lines.