Found problems: 1704
1996 Yugoslav Team Selection Test, Problem 2
Let there be given a set of $1996$ equal circles in the plane, no two of them having common interior points. Prove that there exists a circle touching at most three other circles.
2007 IMO Shortlist, 8
Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
1990 All Soviet Union Mathematical Olympiad, 530
A cube side $100$ is divided into a million unit cubes with faces parallel to the large cube. The edges form a lattice. A prong is any three unit edges with a common vertex. Can we decompose the lattice into prongs with no common edges?
1977 All Soviet Union Mathematical Olympiad, 235
Given a closed broken line without self-intersections in a plane. Not a triple of its vertices belongs to one straight line. Let us call "special" a couple of line's segments if the one's extension intersects another. Prove that there is even number of special pairs.
1966 IMO Longlists, 14
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]
2011 Korea Junior Math Olympiad, 4
For a positive integer $n$, ($n\ge 2$), find the number of sets with $2n + 1$ points $P_0, P_1,..., P_{2n}$ in the coordinate plane satisfying the following as its elements:
- $P_0 = (0, 0),P_{2n}= (n, n)$
- For all $i = 1,2,..., 2n - 1$, line $P_iP_{i+1}$ is parallel to $x$-axis or $y$-axis and its length is $1$.
- Out of $2n$ lines$P_0P_1, P_1P_2,..., P_{2n-1}P_{2n}$, there are exactly $4$ lines that are enclosed in the domain $y \le x$.
1997 Estonia National Olympiad, 4
There are $19$ lines in the plane dividing the plane into exactly $97$ pieces.
(a) Prove that among these pieces there is at least one triangle.
(b) Show that it is indeed possible to place $19$ lines in the above way.
2020 New Zealand MO, 3
There are $13$ marked points on the circumference of a circle with radius $13$. Prove that we can choose three of the marked points which form a triangle with area less than $13$.
2017 QEDMO 15th, 2
Markers in the colors violet, cyan, octarine and gamma were placed on all fields of a $41\times 5$ chessboard. Show that there are four squares of the same color that form the vertices of a rectangle whose edges are parallel to those of the board.
1987 Greece National Olympiad, 1
It is known that diagonals of a square, as well as a regular pentagon, are all equal. Find the bigeest natural $n$ such that a convex $n$-gon has all it's diagonals equal.
1999 Harvard-MIT Mathematics Tournament, 10
If $5$ points are placed in the plane at lattice points (i.e. points $(x,y)$ where $x $and $y$ are both integers) such that no three are collinear, then there are $10$ triangles whose vertices are among these points. What is the minimum possible number of these triangles that have area greater than $1/2$?
2013 Estonia Team Selection Test, 2
For which positive integers $n \ge 3$ is it possible to mark $n$ points of a plane in such a way that, starting from one marked point and moving on each step to the marked point which is the second closest to the current point, one can walk through all the marked points and return to the initial one? For each point, the second closest marked point must be uniquely determined.
1981 Austrian-Polish Competition, 8
The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?
2001 All-Russian Olympiad Regional Round, 9.5
Two points are selected in a convex pentagon. Prove that you can choose a quadrilateral with vertices at the vertices of a pentagon so that both selected points fall into it.
2023 Portugal MO, 6
A rectangular board, where in each square there is a symbol, is said to be [i]magnificent [/i] if, for each line$ L$ and for each pair of columns $C$ and $D$, there is on the board another line $M$ exactly equal to $L$, except in columns $C$ and $D$, where $M$ has symbols different from those of $L$. What is the smallest possible number of rows on a magnificent board with $2023$ columns?
1993 Bundeswettbewerb Mathematik, 1
In a regular nonagon, each vertex is colored either red or green. Three corners of the nonagon determine a triangle. Such a triangle is called [i]red [/i] or [i]green [/i] if all its vertices are red or green if all are green. Prove that for each such coloring of the nonagon there are at least two different ones , that are congruent triangles of the same color.
1993 Bulgaria National Olympiad, 6
Find all natural numbers $n$ for which there exists set $S$ consisting of $n$ points in the plane, satisfying the condition:
For each point $A \in S$ there exist at least three points say $X, Y, Z$ from $S$ such that the segments $AX, AY$ and$ AZ$ have length $1$ (it means that $AX = AY = AZ = 1$).
1988 All Soviet Union Mathematical Olympiad, 483
A polygonal line with a finite number of segments has all its vertices on a parabola. Any two adjacent segments make equal angles with the tangent to the parabola at their point of intersection. One end of the polygonal line is also on the axis of the parabola. Show that the other vertices of the polygonal line are all on the same side of the axis.
I Soros Olympiad 1994-95 (Rus + Ukr), 10.4
There are 1995 segments such that a triangle can be formed from any three of them. Prove that using these $1995 $ segments, it is possible to assemble $664$ acute-angled triangles so that each segment is part of no more than one triangle.
1954 Kurschak Competition, 2
Every planar section of a three-dimensional body $B$ is a disk. Show that B must be a ball.
1994 Tournament Of Towns, (436) 2
Show how to divide space into
(a) congruent tetrahedra,
(b) congruent “equifaced” tetrahedra.
(A tetrahedron is called equifaced if all its faces are congruent triangles.)
(NB Vassiliev)
1987 Tournament Of Towns, (135) 4
We are given tiles in the form of right angled triangles having perpendicular sides of length $1$ cm and $2$ cm. Is it possible to form a square from $20$ such tiles?
( S . Fomin , Leningrad)
1948 Moscow Mathematical Olympiad, 143
On a plane, $n$ straight lines are drawn. Two domains are called [i]adjacent [/i] if they border by a line segment. Prove that the domains into which the plane is divided by these lines can be painted two colors so that no two [i]adjacent [/i] domains are of the same color.
May Olympiad L2 - geometry, 2012.4
Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.
1963 Miklós Schweitzer, 10
Select $ n$ points on a circle independently with uniform distribution. Let $ P_n$ be the probability that the center of the
circle is in the interior of the convex hull of these $ n$ points. Calculate the probabilities $ P_3$ and $ P_4$. [A. Renyi]