Found problems: 1704
1965 Polish MO Finals, 3
$ n > 2 $ points are chosen on a circle and each of them is connected to every other by a segment. Is it possible to draw all of these segments in one sequence, i.e. so that the end of the first segment is the beginning of the second, the end of the second - the beginning of the third, etc., and so that the end of the last segment is the beginning of the first?
2016 Ukraine Team Selection Test, 5
Let $ABC$ be an equilateral triangle of side $1$. There are three grasshoppers sitting in $A$, $B$, $C$. At any point of time for any two grasshoppers separated by a distance $d$ one of them can jump over other one so that distance between them becomes $2kd$, $k,d$ are nonfixed positive integers. Let $M$, $N$ be points on rays $AB$, $AC$ such that $AM=AN=l$, $l$ is fixed positive integer. In a finite number of jumps all of grasshoppers end up sitting inside the triangle $AMN$. Find, in terms of $l$, the number of final positions of the grasshoppers. (Grasshoppers can leave the triangle $AMN$ during their jumps.)
2018 Estonia Team Selection Test, 11
Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied:
1) there is at least $1$ marked point on each side,
2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$
2017 Bulgaria EGMO TST, 1
Prove that every convex polygon has at most one triangulation consisting entirely of acute triangles.
1999 Cono Sur Olympiad, 6
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 .
1967 Swedish Mathematical Competition, 6
The vertices of a triangle are lattice points. There are no lattice points on the sides (apart from the vertices) and $n$ lattice points inside the triangle. Show that its area is $n + \frac12$. Find the formula for the general case where there are also $m$ lattice points on the sides (apart from the vertices).
2015 Middle European Mathematical Olympiad, 2
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.
1984 Swedish Mathematical Competition, 2
The squares in a $3\times 7$ grid are colored either blue or yellow. Consider all $m\times n$ rectangles in this grid, where $m \in \{2,3\}$, $n \in \{2,...,7\}$. Prove that at least one of these rectangles has all four corner squares the same color.
2000 Tuymaada Olympiad, 2
Is it possible to paint the plane in $4$ colors so that inside any circle are the dots of all four colors?
1999 All-Russian Olympiad Regional Round, 10.8
Some natural numbers are marked. It is known that on every a segment of the number line of length $1999$ has a marked number. Prove that there is a pair of marked numbers, one of which is divisible by the other.
1971 Polish MO Finals, 6
A regular tetrahedron with unit edge length is given. Prove that:
(a) There exist four points on the surface $S$ of the tetrahedron, such that the distance from any point of the surface to one of these four points does not exceed $1/2$;
(b) There do not exist three points with this property.
The distance between two points on surface $S$ is defined as the length of the shortest polygonal line going over $S$ and connecting the two points.
1978 Polish MO Finals, 2
In a coordinate plane, consider the set of points with integer cooedinates at least one of which is not divisible by $4$. Prove that these points cannot be partitioned into pairs such that the distance between points in each pair equals $1$.
In other words, an infinite chessboard, whose cells with both coordinates divisible by $4$ are cut out, cannot be tiled by dominoes.
1973 Czech and Slovak Olympiad III A, 6
Consider a square of side of length 50. A polygonal chain $L$ is given in the square such that for every point $P$ of the square there is a point $Q$ of the chain with the property $PQ\le 1.$ Show that the length of $L$ is greater than 1248.
2011 Tournament of Towns, 2
In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is $2011$, prove that the sides of the rectangular box are parallel to the coordinate axes.
1999 Tournament Of Towns, 4
A black unit equilateral triangle is drawn on the plane. How can we place nine tiles, each a unit equilateral triangle, on the plane so that they do not overlap, and each tile covers at least one interior point of the black triangle?
(Folklore)
1998 Singapore MO Open, 2
Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.
1991 Mexico National Olympiad, 6
Given an $n$-gon ($n\ge 4$), consider a set $T$ of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that $T$ contains at most $n$ triangles.
2023 Sharygin Geometry Olympiad, 9.2
Can a regular triangle be placed inside a regular hexagon in such a way that all vertices of the triangle were seen from each vertex of the hexagon? (Point $A$ is seen from $B$, if the segment $AB$ dots not contain internal points of the triangle.)
2020 Olympic Revenge, 5
Let $n$ be a positive integer. Given $n$ points in the plane, prove that it is possible to draw an angle with measure $\frac{2\pi}{n}$ with vertex as each one of the given points, such that any point in the plane is covered by at least one of the angles.
1971 Swedish Mathematical Competition, 2
An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.
1968 Polish MO Finals, 6
Consider a set of $n > 3$ points in the plane, no three of which are collinear, and a natural number $k < n$. Prove the following statements:
(a) If $k \le \frac{n}{2}$, then each point can be connected with at least k other points by segments so that no three segments form a triangle.
(b) If $k \ge \frac{n}{2}$, and each point is connected with at least k other points by segments, then some three segments form a triangle.
1982 Poland - Second Round, 2
The plane is covered with circles in such a way that the center of each circle does not belong to any other circle. Prove that each point of the plane belongs to at most five circles.
1968 Poland - Second Round, 6
On the plane are chosen $n \ge 3$ points, not all on the same line. Drawing all lines passing through two of these points one obtains k different lines. Prove that $k \ge n$.
2021 Poland - Second Round, 5
Find the largest positive integer $n$ with the following property: there are rectangles $A_1, ... , A_n$ and $B_1,... , B_n,$ on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles $A_i$ and $B_i$ are disjoint for all $i \in \{1,..., n\}$, but the rectangles $A_i$ and $B_j$ have a common point for all $i, j \in \{1,..., n\}$, $i \ne j$.
[i]Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices [/i]
2019 May Olympiad, 5
We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set?
[hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno
de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]