Found problems: 1704
1974 Dutch Mathematical Olympiad, 5
For every $n \in N$, is it possible to make a figure consisting of $n+1$ points, where $n$ points lie on one line and one point is not on that line, so that each pair of those points is an integer distance from each other?
1999 Bundeswettbewerb Mathematik, 4
It is known that there are polyhedrons whose faces are more numbered than the vertices. Find the smallest number of triangular faces that such a polyhedron can have.
2024 Belarusian National Olympiad, 11.8
Projector emits rays in space. Consider all acute angles between the rays. It is known that no matter what ray we remove, the number of acute angles decreases by exactly $2$
What is the maximal number of rays the projector can emit?
[i]M. Karpuk, E. Barabanov[/i]
2019 New Zealand MO, 5
An equilateral triangle is partitioned into smaller equilateral triangular pieces. Prove that two of the pieces are the same size.
1983 Austrian-Polish Competition, 6
Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$.
1951 Polish MO Finals, 1
A beam of length $ a $ is suspended horizontally with its ends on two parallel ropes equal $ b $. We turn the beam through an angle $ \varphi $ around a vertical axis passing through the center of the beam. By how much will the beam rise?
1986 Polish MO Finals, 1
A square of side $1$ is covered with $m^2$ rectangles.
Show that there is a rectangle with perimeter at least $\frac{4}{m}$.
2000 Czech And Slovak Olympiad IIIA, 3
In the plane are given $2000$ congruent triangles of area $1$, which are all images of one triangle under translations. Each of these triangles contains the centroid of every other triangle. Prove that the union of these triangles has area less than $22/9$.
2004 Estonia National Olympiad, 5
Let $n$ and $c$ be coprime positive integers. For any integer $i$, denote by $i' $ the remainder of division of product $ci$ by $n$. Let $A_o.A_1,A_2,...,A_{n-1}$ be a regular $n$-gon. Prove that
a) if $A_iA_j \parallel A_kA_i$ then $A_{i'}A_{j'} \parallel A_{k'}A_{i'}$
b) if $A_iA_j \perp A_kA_l$ then $A_{i'}A_{j'} \perp A_{k'}A_{l'}$
1994 Poland - Second Round, 4
Each vertex of a cube is assigned $1$ or $-1$. Each face is assigned the product of the four numbers at its vertices. Determine all possible values that can be obtained as the sum of all the $14$ assigned numbers.
Mathley 2014-15, 1
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.
1989 Greece Junior Math Olympiad, 2
How many paths are there from $A$ to $B$ that consist of $5$ horizontal segments and $5$ vertical segments of length $1$ each? (see figure)
[img]https://cdn.artofproblemsolving.com/attachments/4/2/5b476ca2a232fc67fb2e2f6bb06111cab60692.png[/img]
2009 All-Russian Olympiad Regional Round, 11.5
We drew several straight lines on the plane and marked all of them intersection points. How many lines could be drawn? if one point is marked on one of the drawn lines, on the other - three, and on the third - five? Find all possible options and prove that there are no others.
1997 Tournament Of Towns, (546) 7
Several strips and a circle of radius $1$ are drawn on the plane. The sum of the widths of the strips is $100$. Prove that one can translate each strip parallel to itself so that together they cover the circle.
(M Smurov )
2012 Cuba MO, 4
With $21$ pieces, some white and some black, a rectangle is formed of $3 \times 7$. Prove that there are always four pieces of the same color located at the vertices of a rectangle.
2016 Saint Petersburg Mathematical Olympiad, 6
The circle contains a closed $100$-part broken line, such that no three segments pass through one point. All its corners are obtuse, and their sum in degrees is divided by $720$. Prove that this broken line has an odd number of self-intersection points.
2012 Sharygin Geometry Olympiad, 24
Given are $n$ $(n > 2)$ points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?
2006 BAMO, 1
All the chairs in a classroom are arranged in a square $n\times n$ array (in other words, $n$ columns and $n$ rows), and every chair is occupied by a student. The teacher decides to rearrange the students according to the following two rules:
(a) Every student must move to a new chair.
(b) A student can only move to an adjacent chair in the same row or to an adjacent chair in the same
column. In other words, each student can move only one chair horizontally or vertically.
(Note that the rules above allow two students in adjacent chairs to exchange places.)
Show that this procedure can be done if $n$ is even, and cannot be done if $n$ is odd.
2002 All-Russian Olympiad Regional Round, 11.6
There are $n > 1$ points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?
2014 Belarusian National Olympiad, 8
An $n\times n$ square is divided into $n^2$ unit cells. Is it possible to cover this square with some layers of 4-cell figures of the following shape [img]https://cdn.artofproblemsolving.com/attachments/5/7/d42a8011ec4c5c91c337296d8033d412fade5c.png[/img](i.e. each cell of the square must be covered with the same number of these figures) if
a) $n=6$?
b) $n=7$?
(The sides of each figure must coincide with the sides of the cells; the figures may be rotated and turned over, but none of them can go beyond the bounds of the square.)
1970 Swedish Mathematical Competition, 2
$6$ open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all $6$ disks.
2012 Online Math Open Problems, 29
In the Cartesian plane, let $S_{i,j} = \{(x,y)\mid i \le x \le j\}$. For $i=0,1,\ldots,2012$, color $S_{i,i+1}$ pink if $i$ is even and gray if $i$ is odd. For a convex polygon $P$ in the plane, let $d(P)$ denote its pink density, i.e. the fraction of its total area that is pink. Call a polygon $P$ [i]pinxtreme[/i] if it lies completely in the region $S_{0,2013}$ and has at least one vertex on each of the lines $x=0$ and $x=2013$. Given that the minimum value of $d(P)$ over all non-degenerate convex pinxtreme polygons $P$ in the plane can be expressed in the form $\frac{(1+\sqrt{p})^2}{q^2}$ for positive integers $p,q$, find $p+q$.
[i]Victor Wang.[/i]
1983 Spain Mathematical Olympiad, 3
A semicircle of radius $r$ is divided into $n + 1$ equal parts and any point $k$ of the division with the ends of the semicircle forms a triangle $A_k$. Calculate the limit, as $n$ tends to infinity, of the arithmetic mean of the areas of the triangles.
1975 Polish MO Finals, 2
On the surface of a regular tetrahedron of edge length $1$ are given finitely many segments such that every two vertices of the tetrahedron can be joined by a polygonal line consisting of given segments. Can the sum of the lengths of the given segments be less than $1+\sqrt3 $?
2012 Denmark MO - Mohr Contest, 2
It is known about a given rectangle that it can be divided into nine squares which are situated relative to each other as shown. The black rectangle has side length $1$. Are there more than one possibility for the side lengths of the rectangle?
[img]https://cdn.artofproblemsolving.com/attachments/1/0/af6bc5b867541c04586e4b03db0a7f97f8fe87.png[/img]