Found problems: 1704
2023 Serbia Team Selection Test, P6
There are $n^2$ segments in the plane (read walls), no two of which are parallel or intersecting.
Prove that there are at least $n$ points in the plane such that no two of them see each other (meaning there is a wall separating them).
1974 Spain Mathematical Olympiad, 1
It is known that a regular dodecahedron is a regular polyhedron with $12$ faces of equal pentagons and concurring $3$ edges in each vertex. It is requested to calculate, reasonably,
a) the number of vertices,
b) the number of edges,
c) the number of diagonals of all faces,
d) the number of line segments determined for every two vertices,
d) the number of diagonals of the dodecahedron.
TNO 2023 Senior, 6
The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).
2013 Saudi Arabia BMO TST, 7
Ayman wants to color the cells of a $50 \times 50$ chessboard into black and white so that each $2 \times 3$ or $3 \times 2$ rectangle contains an even number of white cells. Determine the number of ways Ayman can color the chessboard.
2013 Tournament of Towns, 3
There is a $19\times19$ board. Is it possible to mark some $1\times 1$ squares so that each of $10\times 10$ squares contain different number of marked squares?
2011 QEDMO 8th, 5
$9$ points are given in the interior of the unit square.
Prove there exists a triangle of area $\le \frac18$ whose vertices are three of the points.
2007 China Team Selection Test, 3
Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$
2024 India IMOTC, 24
There are $n > 1$ distinct points marked in the plane. Prove that there exists a set of circles $\mathcal C$ such that
[color=#FFFFFF]___[/color]$\bullet$ Each circle in $\mathcal C$ has unit radius.
[color=#FFFFFF]___[/color]$\bullet$ Every marked point lies in the (strict) interior of some circle in $\mathcal C$.
[color=#FFFFFF]___[/color]$\bullet$ There are less than $0.3n$ pairs of circles in $\mathcal C$ that intersect in exactly $2$ points.
[i]Note: Weaker results with $\it{0.3n}$ replaced by $\it{cn}$ may be awarded points depending on the value of the constant $\it{c > 0.3}$.[/i]
[i]Proposed by Siddharth Choppara, Archit Manas, Ananda Bhaduri, Manu Param[/i]
2006 Junior Balkan Team Selection Tests - Romania, 2
In a plane $5$ points are given such that all triangles having vertices at these points are of area not greater than $1$. Show that there exists a trapezoid which contains all point in the interior (or on the sides) and having the area not exceeding $3$.
2016 Brazil National Olympiad, 2
Find the smallest number \(n\) such that any set of \(n\) ponts in a Cartesian plan, all of them with integer coordinates, contains two poitns such that the square of its mutual distance is a multiple of \(2016\).
2015 Sharygin Geometry Olympiad, P12
Find the maximal number of discs which can be disposed on the plane so that each two of them have a common point and no three have it
1995 All-Russian Olympiad Regional Round, 10.7
$N^3$ unit cubes are made into beads by drilling a hole through them along a diagonal, put on a string and binded. Thus the cubes can move freely in space as long as the vertices of two neighboring cubes (including the first and last one) are touching. For which $N$ is it possible to build a cube of edge $N$ using these cubes?
1997 All-Russian Olympiad Regional Round, 11.7
Are there convex $n$-gonal ($n \ge 4$) and triangular pyramids such that the four trihedral angles of the $n$-gonal pyramid are equal trihedral angles of a triangular pyramid?
[hide=original wording] Существуют ли выпуклая n-угольная (n>= 4) и треугольная пирамиды такие, что четыре трехгранных угла n-угольной пирамиды равны трехгранным углам треугольной пирамиды?[/hide]
2024 Junior Balkan Team Selection Tests - Moldova, 3
Let $M$ be a set of 999 points in the plane with the property: For any 3 distinct points in $M$ we can choose two of them, such that the distance between them is less than $1$.
a)Prove that there exists a disc of radius not greater than 1 that covers at least 500 points in $M$.
b)Is it true that there always exists a disc of radius not greater than 1 that covers at least 501 points in $M$?
1978 Poland - Second Round, 4
Three different points were randomly selected from the vertices of the regular $2n$-gon. Let $ p_n $ be the probability of the event that the triangle with vertices at the selected points is acute-angled. Calculate $ \lim_{n\to \infty} p_n $.
Attention. We assume that all choices of three different points are equally likely.
2015 Romania Masters in Mathematics, 6
Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every set $C$ of $4n$ points in the interior of the unit square $U$, there exists a rectangle $T$ contained in $U$ such that
$\bullet$ the sides of $T$ are parallel to the sides of $U$;
$\bullet$ the interior of $T$ contains exactly one point of $C$;
$\bullet$ the area of $T$ is at least $\mu$.
1987 Polish MO Finals, 6
A plane is tiled with regular hexagons of side $1$. $A$ is a fixed hexagon vertex.
Find the number of paths $P$ such that:
(1) one endpoint of $P$ is $A$,
(2) the other endpoint of $P$ is a hexagon vertex,
(3) $P$ lies along hexagon edges,
(4) $P$ has length $60$, and
(5) there is no shorter path along hexagon edges from $A$ to the other endpoint of $P$.
2012 Tournament of Towns, 2
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.
2006 Estonia National Olympiad, 2
Prove that the circle with radius $2$ can be completely covered with $7$ unit circles
1978 Austrian-Polish Competition, 9
In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.
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 $\ell_4, \ell_5, \ell_6$.
2016 China Team Selection Test, 5
Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other.
Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.
2010 Belarus Team Selection Test, 6.3
A $50 \times 50$ square board is tiled by the tetrominoes of the following three types:
[img]https://cdn.artofproblemsolving.com/attachments/2/9/62c0bce6356ea3edd8a2ebfe0269559b7527f1.png[/img]
Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling.
(Folklore)
2024 239 Open Mathematical Olympiad, 2
There are $2n$ points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose $n$ drawn segments having no common ends?
1998 China Team Selection Test, 2
Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.