Found problems: 1704
2011 Rioplatense Mathematical Olympiad, Level 3, 5
A [i]form [/i] is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, ... , m_n$ satisying $m_1\ge ... \ge m_n$. An [i]angle [/i] in a [i]form [/i] consists of a box $v$ and of all the boxes to the right of $v$ and all the boxes above $v$. The size of a [i]form [/i] of an [i]angle [/i] is the number of boxes it contains. Find the maximum number of [i]angles [/i] of size $11$ in a form of size $400$.
[url=http://www.oma.org.ar/enunciados/omr20.htm]source[/url]
2020 Princeton University Math Competition, 11
Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.
2018 Romania Team Selection Tests, 3
Divide the plane into $1$x$1$ squares formed by the lattice points. Let$S$ be the set-theoretic union of a finite number of such cells, and let $a$ be a positive real number less than or equal to 1/4.Show that S can be covered by a finite number of squares satisfying the following three conditions:
1) Each square in the cover is an array of $1$x$1$ cells
2) The squares in the cover have pairwise disjoint interios and
3)For each square $Q$ in the cover the ratio of the area $S \cap Q$ to the area of Q is at least $a$ and at most
$a {(\lfloor a^{-1/2} \rfloor)} ^2$
2012 JBMO ShortLists, 3
In a circle of diameter $1$ consider $65$ points, no three of them collinear. Prove that there exist three among these points which are the vertices of a triangle with area less than or equal to $\frac{1}{72}$.
2019 Istmo Centroamericano MO, 5
Gabriel plays to draw triangles using the vertices of a regular polygon with $2019$ sides, following these rules:
(i) The vertices used by each triangle must not have been previously used.
(ii) The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn.
If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.
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]
1959 Poland - Second Round, 5
In the plane, $ n \geq 3 $ segments are placed in such a way that every $ 3 $ of them have a common point. Prove that there is a common point for all the segments.
2018 NZMOC Camp Selection Problems, 7
Let $N$ be the number of ways to colour each cell in a $2 \times 50$ rectangle either red or blue such that each $2 \times 2$ block contains at least one blue cell. Show that $N$ is a multiple of $3^{25}$, but not a multiple of $3^{26}$
2003 All-Russian Olympiad Regional Round, 9.8
Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.
1990 ITAMO, 1
A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.
1981 Brazil National Olympiad, 4
A graph has $100$ points. Given any four points, there is one joined to the other three. Show that one point must be joined to all $99$ other points. What is the smallest number possible of such points (that are joined to all the others)?
1996 Portugal MO, 6
In a regular polygon with $134$ sides, $67$ diagonals are drawn so that exactly one diagonal emerges from each vertex. We call the [i]length[/i] of a diagonal the number of sides of the polygon included between the vertices of the diagonal and which is less than or equal to $67$. If we order the [i]lengths [/i] of the diagonals in ascending order, we obtain a succession of $67$ numbers $(d_1,d_2,...,d_{67})$. It will be possible to draw diagonals such that
a) $(d_1,d_2,...,d_{67})=\underbrace{2 ... 2}_{6},\underbrace{3 ... 3}_{61}$ ?
b) $(d_1,d_2,...,d_{67}) =\underbrace{3 ... 3}_{8},\underbrace{6 ... 6}_{55}.\underbrace{8 ... 8}_{4} $ ?
1945 Moscow Mathematical Olympiad, 094
Prove that it is impossible to divide a scalene triangle into two equal triangles.
2012 QEDMO 11th, 5
Let $n$ be a natural number and $L = Z^2$ the set of points on the plane with integer coordinates. Every point in $L$ is colored now in one of the colors red or green. Show that there are $n$ different points $x_1,...,x_n \in L$ all of which have the same color and whose center of gravity is also in $L$ and is of the same color.
2019 Tuymaada Olympiad, 5
Is it possible to draw in the plane the graph presented in the figure so that all the vertices are different points and all the edges are unit segments? (The segments can intersect at points different from vertices.)
1966 IMO Longlists, 52
A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$
2013 China Northern MO, 1
Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.
1974 Chisinau City MO, 80
Each side face of a regular hexagonal prism is colored in one of three colors (for example, red, yellow, blue), and the adjacent prism faces have different colors. In how many different ways can the edges of the prism be colored (using all three colors is optional)?
2011 Romanian Masters In Mathematics, 2
For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$.
(We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.)
[i](United Kingdom) Luke Betts[/i]
1965 IMO Shortlist, 6
In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.
2015 Caucasus Mathematical Olympiad, 3
The workers laid a floor of size $n \times n$ with tiles of two types: $2 \times 2$ and $3 \times 1$.
It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen?
(You can’t cut tiles and also put them on top of each other.)
1999 All-Russian Olympiad Regional Round, 9.5
All cells of the checkered plane are painted in $5$ colors so that in any figure of the species [img]https://cdn.artofproblemsolving.com/attachments/f/f/49b8d6db20a7e9cca7420e4b51112656e37e81.png[/img] all colors are different. Prove that in any figure of the species $ \begin{tabular}{ | l | c| c | c | r| } \hline & & & &\\ \hline \end{tabular}$, all colors are different..
1984 All Soviet Union Mathematical Olympiad, 394
Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.
2001 German National Olympiad, 4
In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn.
[img]https://cdn.artofproblemsolving.com/attachments/0/5/33795820e0335686b06255180af698e536a9be.png[/img]
1994 Tournament Of Towns, (430) 7
The figure $F$ is the intersection of $N$ circles (they may have different radii). Find the maximal number of curvilinear “sides” which $F$ can have. Curvilinear sides of $F$ are the arcs (of the given circumferences) that constitute the boundary of $F$. (Their ends are the “vertices” of $F$ - the points of intersection of given circumferences that lie on the boundary of $F$.)
(N Brodsky)