Found problems: 121
Let $A_1,\ldots,A_n$ be different collinear points. Every point is dyed by one of four colors and every of these colors is used at least once. Show that there is a line segment where two colors are used exactly once and the other two are used at least once.
Find all positive integers $n \ge 4$ for which there are $n$ points in general position on the plane such that an arbitrary triangle with vertices belonging to the convex hull of these $n$ points, containing exactly one of $n - 3$ points inside remained.
Let $A_1,A_2,...,A_n$ be $n \ge 2$ distinct points on a circle.
Find the number of colorings of these points with $p \ge 2$ colors such that every two adjacent points receive different colors
Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$.
A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$.
A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$.
Now put a pawn on $(0, 0)$. You can make a ( nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B.
Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.
In the plane there are two disjoint sets $ A $ and $ B $, each of which consists of $ n $ points, and no three points of the set $ A \cup B $ lie on one straight line. Prove that there is a set of $ n $ disjoint closed segments, each of which has one end in the set $ A $ and the other in the set $ B $.
In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?
Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.
$ n > 2 $ points are chosen on a circle and each of them is connected to every other by a segment. Is it possible to draw all of these segments in one sequence, i.e. so that the end of the first segment is the beginning of the second, the end of the second - the beginning of the third, etc., and so that the end of the last segment is the beginning of the first?
(a) $10$ points dividing a circle into $10$ equal arcs are connected in pairs by $5$ chords.
Is it necessary that two of these chords are of equal length?
(b) $20$ points dividing a circle into $20$ equal arcs are connected in pairs by $10$ chords.
Prove that among these $10$ chords there are two chords of equal length.
(VV Proizvolov, Moscow)
In the plane there are $2014$ plotted points, such that no $3$ are collinear. For each pair of plotted points, draw the line that passes through them. prove that for every three of marked points there are always two that are separated by an amount odd number of lines.
Assuming that each point of a straight line is painted red or blue, arbitrarily, show that it is always possible to choose three points $A, B$ and $C$ in such a way straight, that are painted the same color and that: $$\frac{AB}{1}=\frac{BC}{2}=\frac{AC}{3}.$$
On a circle we consider a set $M$ consisting of $n$ ($n \ge 3$) points, of which only one is colored red. Determine of which polygons inscribed in a circle having the vertices in the set $M$ are more: those that contain the red dot or those that do not contain those points? How many more are there than others?
For what positive numbers $m$ and $n$ do there exist points $A_1, ..., Am$ and $B_1 ..., B_n$ in the plane such that, for any point $P$, the equation $$|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2 $$ holds true?
At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw score and zero points for the defeat. At the end of the tournament it turned out that the sum of the winning points of all teams was $50$.
(a) How many teams participated in this tournament?
(b) How big is the difference between the team with the highest number and the number of points won?
Let $M$ be the set of integer coordinate points situated on the line $d$ of real numbers. We color the elements of M in black or white. Show that at least one of the following statements is true:
(a) there exists a finite subset $F \subset M$ and a point $M \in d$ so that the elements of the set $M - F$ that are lying on one of the rays determined by $M$ on $d$ are all white, and the elements of $M - F$ that are situated on the opposite ray are all black,
(b) there exists an infinite subset $S \subset M$ and a point $T \in d$ so that for each $A \in S$ the reflection of A about $T$ belongs to $S$ and has the same color as $A$
World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$
Ten points on a plane a such that any four of them lie on the boundary of some square. Is obligatory true that all ten points lie on the boundary of some square?
Mikko takes a multiple choice test with ten questions. His only goal is to pass the test, and this requires seven points. A correct answer is worth one point, and answering wrong results in the deduction of one point. Mikko knows for sure that he knows the correct answer in the six first questions. For the rest, he estimates that he can give the correct answer to each problem with probability $p, 0 < p < 1$. How many questions Mikko should try?
On the plane $n+1$ points are marked, no three of which lie on one straight line. For what natural $k$ can they be connected by segments so that for any $n$ marked points there are exactly $k$ segments with ends at these points?
On the lines $AB$ are located $2009$ different points that do not belong to the segment $[AB]$. Prove that the sum of the distances from point $A$ to these points is not equal to the sum of the distances from point $B$ to these points.
Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)
Do there exist $100$ straight lines on a plane such that they intersect each other in exactly $1999$ points?
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 $d$.
In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.
The $n$ participants of a tournament are numbered with $0$ through $n - 1$. At the end of the tournament it turned out that for every team, numbered with $s$ and having $t$ points, there are exactly $t$ teams having $s$ points each. Determine all possibilities for the final score list.