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.

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Found problems: 1704

2012 QEDMO 11th, 8
Tags: combinatorial geometry , combinatorics , point
Prove that there are $2012$ points in the plane, none of which are three on one straight line and in pairs have integer distances .

1993 Tournament Of Towns, (366) 5
Tags: combinatorial geometry , similar triangles , geometry
A paper triangle with the angles $20^o$, $20^o$ and $140^o$ is cut into two triangles by the bisector of one of its angles. Then one of these triangles is cut into two by its bisector, and so on. Prove that it is impossible to get a triangle similar to the initial one. (AI Galochkin)

2008 All-Russian Olympiad, 8
Tags: induction , symmetry , modular arithmetic , combinatorial geometry , combinatorics
In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

2004 Estonia National Olympiad, 3
Tags: combinatorics , combinatorial geometry
From $25$ points in a plane, both of whose coordinates are integers of the set $\{-2,-1, 0, 1, 2\}$, some $17$ points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.

2000 BAMO, 4
Tags: combinatorics , combinatorial geometry , distance , point
Prove that there exists a set $S$ of $3^{1000}$ points in the plane such that for each point $P$ in $S$, there are at least $2000$ points in $S$ whose distance to $P$ is exactly $1$ inch.

1994 Chile National Olympiad, 4
Tags: combinatorial geometry , combinatorics , circles , 3d geometry , sphere , geometry
Consider a box of dimensions $10$ cm $\times 16$ cm $\times 1$ cm. Determine the maximum number of balls of diameter $ 1$ cm that the box can contain.

1999 Cono Sur Olympiad, 6
Tags: combinatorial geometry , geometry , rotation
An ant walks across the floor of a circular path of radius $r$ and moves in a straight line, but sometimes stops. Each time it stops, before resuming the march, it rotates $60^o$ alternating the direction (if the last time it turned $60^o$ to its right, the next one does it $60^o$ to its left, and vice versa). Find the maximum possible length of the path the ant goes through. Prove that the length found is, in fact, as long as possible. Figure: turn $60^o$ to the right .

2010 Chile National Olympiad, 6
Tags: geometry , combinatorics , combinatorial geometry
Prove that in the interior of an equilateral triangle with side $a$ you can put a finite number of equal circles that do not overlap, with radius $r = \frac{a}{2010}$, so that the sum of their areas is greater than $\frac{17\sqrt3}{80}$ a$^2$.

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 $x$.

2010 Moldova Team Selection Test, 4
Tags: induction , symmetry , modular arithmetic , combinatorial geometry , combinatorics
In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

2015 Middle European Mathematical Olympiad, 2
Tags: combinatorics , polygon , combinatorial geometry , diagonal
Let $n\ge 3$ be an integer. An [i]inner diagonal[/i] of a [i]simple $n$-gon[/i] is a diagonal that is contained in the $n$-gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$-gon $P$ and by $D(n)$ the least possible value of $D(Q)$, where $Q$ is a simple $n$-gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P)=D(n)$. [i]Remark:[/i] A simple $n$-gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.

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 $$S=\{v \times w: v, w \in S\}?$$

2019 Costa Rica - Final Round, 1
Tags: combinatorics , geometry , combinatorial geometry
In a faraway place in the Universe, a villain has a medal with special powers and wants to hide it so that no one else can use it. For this, the villain hides it in a vertex of a regular polygon with $2019$ sides. Olcoman, the savior of the Olcomita people, wants to get the medal to restore peace in the Universe, for which you have to pay $1000$ olcolones for each time he makes the following move: on each turn he chooses a vertex of the polygon, which turns green if the medal is on it or in one of the four vertices closest to it, or otherwise red. Find the fewest olcolones Olcoman needs to determine with certainty the position of the medal.

1966 IMO Shortlist, 14
Tags: combinatorics , combinatorial geometry , counting , circles , arrangement
What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]

1974 Kurschak Competition, 2
Tags: combinatorics , combinatorial geometry
$S_n$ is a square side $\frac{1}{n}$. Find the smallest $k$ such that the squares $S_1, S_2,S_3, ...$ can be put into a square side $k$ without overlapping.

2022 Spain Mathematical Olympiad, 1
Tags: combinatorics , combinatorial geometry
The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. Each of the thirteen marked points is assigned a color, green or red. Prove that there are always three points of the same color, which are the vertices of an equilateral triangle.

2010 Oral Moscow Geometry Olympiad, 1
Tags: geometry , combinatorial geometry , combinatorics , cyclic , polygon , convex polygon
Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

2001 All-Russian Olympiad Regional Round, 10.1
Tags: algebra , geometry , combinatorial geometry
The lengths of the sides of the polygon are $a_1$, $a_2$,. $..$ ,$a_n$. The square trinomial $f(x)$ is such that $f(a_1) = f(a_2 +...+ a_n)$. Prove that if $A$ is the sum of the lengths of several sides of a polygon, $B$ is the sum of the lengths of its remaining sides, then $f(A) = f(B)$.

2022 Canada National Olympiad, 4
Tags: combinatorics , combinatorial geometry
Call a set of $n$ lines [i]good[/i] if no $3$ lines are concurrent. These $n$ lines divide the Euclidean plane into regions (possible unbounded). A [i]coloring[/i] is an assignment of two colors to each region, one from the set $\{A_1, A_2\}$ and the other from $\{B_1, B_2, B_3\}$, such that no two adjacent regions (adjacent meaning sharing an edge) have the same $A_i$ color or the same $B_i$ color, and there is a region colored $A_i, B_j$ for any combination of $A_i, B_j$. A number $n$ is [i]colourable[/i] if there is a coloring for any set of $n$ good lines. Find all colourable $n$.

2012 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: combinatorial geometry , combinatorics , rectangle , minimum
A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?

1997 Dutch Mathematical Olympiad, 4
Tags: octahedron , coloring , combinatorial geometry , combinatorics
We look at an octahedron, a regular octahedron, having painted one of the side surfaces red and the other seven surfaces blue. We throw the octahedron like a die. The surface that comes up is painted: if it is red it is painted blue and if it is blue it is painted red. Then we throw the octahedron again and paint it again according to the above rule. In total we throw the octahedron $10$ times. How many different octahedra can we get after finishing the $10$th time? [i]Two octahedra are different if they cannot be converted into each other by rotation.[/i]

1985 Tournament Of Towns, (096) 5
Tags: geometry , rectangle , combinatorics , combinatorial geometry , projection , parallelepiped , 3d geometry
A square is divided into rectangles. A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$. (a) Prove that every two rectangles in such a division are members of a certain "chain". (b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") . (A.I . Golberg, V.A. Gurevich)

1993 IMO Shortlist, 2
Tags: algebra , geometry , point set , distance , combinatorial geometry
Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$

1999 All-Russian Olympiad, 6
Tags: rotation , combinatorial geometry , geometry
Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.

1989 Putnam, A5
Tags: geometry , combinatorics , combinatorial geometry
Show that we can find $\alpha>0$ such that, given any point $P$ inside a regular $2n+1$-gon which is inscribed in a circle radius $1$, we can find two vertices of the polygon whose distance from $P$ differ by less than $\frac1n-\frac\alpha{n^3}$.