Found problems: 1704
2016 Saudi Arabia IMO TST, 1
On the Cartesian coordinate system $Oxy$, consider a sequence of points $A_n(x_n, y_n)$ in which $(x_n)^{\infty}_{n=1}$,$(y_n)^{\infty}_{n=1}$ are two sequences of positive numbers satisfing the following conditions:
$$x_{n+1} =\sqrt{\frac{x_n^2+x_{n+2}^2}{2}}, y_{n+1} =\big( \frac{\sqrt{y_n}+\sqrt{y_{n+2}}}{2} \big)^2 \,\, \forall n \ge 1 $$ Suppose that $O, A_1, A_{2016}$ belong to a line $d$ and $A_1, A_{2016}$ are distinct.
Prove that all the points $A_2, A_3,. .. , A_{2015}$ lie on one side of $d$.
2015 Tournament of Towns, 2
A $10 \times 10$ square on a grid is split by $80$ unit grid segments into $20$ polygons of equal area (no one of these segments belongs to the boundary of the square). Prove that all polygons are congruent.
[i]($6$ points)[/i]
1984 Tournament Of Towns, (072) 3
On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely?
(V.E. Kolosov)
1971 IMO Longlists, 8
Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.
2010 China Northern MO, 4
As shown in the figure, chess pieces are placed at the intersection points of the $64$ grid lines of the $7\times 7$ grid table. At most $1$ piece is placed at each point, and a total of $k$ left chess pieces are placed. No matter how they are placed, there will always be $4$ chess pieces, and the grid in which they are located the points form the four vertices of a rectangle (the sides of the rectangle are parallel to the grid lines). Try to find the minimum value of $k$.
[img]https://cdn.artofproblemsolving.com/attachments/5/b/23a79f43d3f4c9aade1ba9eaa7a282c3b3b86f.png[/img]
2005 Switzerland - Final Round, 5
Tweaking a convex $n$-gon means the following: choose two adjacent sides $AB$ and $BC$ and replaces them with the line segment $AM$, $MN$, $NC$, where $M \in AB$ and $N \in BC$ are arbitrary points inside these segments. In other words, you cut off a corner and get an $(n+1)$-corner.
Starting from a regular hexagon $P_6$ with area $1$, by continuous Tweaks a sequence $P_6,P_7,P_8, ...$ convex polygons. Show that Area of $​​P_n$ for all $n\ge 6$ greater than $\frac1 2$ is, regardless of how tweaks takes place.
1984 Tournament Of Towns, (068) T2
A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner?
(Muscovite folklore)
2020 Simon Marais Mathematics Competition, A1
There are $1001$ points in the plane such that no three are collinear. The points are joined by $1001$ line segments such that each point is an endpoint of exactly two of the line segments.
Prove that there does not exist a straight line in the plane that intersects each of the $1001$ segments in an interior point.
[i]An interior point of a line segment is a point of the line segment that is not one of the two endpoints.[/i]
2000 All-Russian Olympiad Regional Round, 11.2
The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?
2015 Ukraine Team Selection Test, 9
The set $M$ consists of $n$ points on the plane and satisfies the conditions:
$\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon,
$\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon.
Find the smallest possible value that $n$ can take .
2018 Costa Rica - Final Round, 1
There are $10$ points on a circle and all possible segments are drawn on the which two of these points are the endpoints. Determine the probability that selecting two segments randomly, they intersect at some point (it could be on the circumference).
IV Soros Olympiad 1997 - 98 (Russia), 10.5
In the lower left corner of the square $7 \times 7$ board there is a king. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different ways can the king get to the upper right corner of the board if he is prohibited from visiting the central square?
Novosibirsk Oral Geo Oly IX, 2022.2
Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?
2015 Swedish Mathematical Competition, 6
Axel and Berta play the following games: On a board are a number of positive integers. One move consists of a player exchanging a number $x$ on the board for two positive integers y and $z$ (not necessarily different), such that $y + z = x$. The game ends when the numbers on the board are relatively coprime in pairs. The player who made the last move has then lost the game. At the beginning of the game, only the number $2015$ is on the board. The two players make do their moves in turn and Berta begins. One of the players has a winning strategy. Who, and why?
1966 IMO Shortlist, 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]
1971 Poland - Second Round, 3
There are 6 lines in space, of which no 3 are parallel, no 3 pass through the same point, and no 3 are contained in the same plane. Prove that among these 6 lines there are 3 mutually oblique lines.
2003 All-Russian Olympiad Regional Round, 10.4
On the plane we mark $n$ ($n > 2$) straight lines passing through one point $O$ in such a way that for any two of them there is a marked straight line that bisects one of the pairs of vertical angles, formed by these straight lines. Prove that the drawn straight lines divide full angle into equal parts.
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]
1953 Moscow Mathematical Olympiad, 251
On a circle, distinct points $A_1, ... , A_{16}$ are chosen. Consider all possible convex polygons all of whose vertices are among $A_1, ... , A_{16}$ . These polygons are divided into $2$ groups, the first group comprising all polygons with $A_1$ as a vertex, the second group comprising the remaining polygons. Which group is more numerous?
1969 Polish MO Finals, 5
For which values of n does there exist a polyhedron having $n$ edges?
2012 CHMMC Spring, 3
Three different faces of a regular dodecahedron are selected at random and painted. What is the probability that there is at least one pair of painted faces that share an edge?
1995 Denmark MO - Mohr Contest, 5
In the plane, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.
Novosibirsk Oral Geo Oly IX, 2022.5
Prove that any triangle can be divided into $22$ triangles, each of which has an angle of $22^o$, and another $23$ triangles, each of which has an angle of $23^o$.
1986 IMO Shortlist, 11
Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$
[i]Simplified version.[/i]
Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$
2021 Israel TST, 2
Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$.
Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.