Found problems: 47
It is allowed to perform the following transformations in the plane with any integers $a$:
(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,
(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.
Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:
a) Vertices of a square,
b) Vertices of a rectangle with unequal side lengths?
A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.
We call a set of squares with sides parallel to the coordinate axes and vertices with integer coordinates friendly if any two of them have exactly two points in common. We consider friendly sets in which each of the squares has sides of length $n$. Determine the largest possible number of squares in such a friendly set.
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 $A$
A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).
Starting at $(1,1)$, a stone is moved in the coordinate plane according to the following rules:
(i) From any point $(a,b)$, the stone can move to $(2a,b)$ or $(a,2b)$.
(ii) From any point $(a,b)$, the stone can move to $(a-b,b)$ if $a > b$, or to $(a,b-a)$ if $a < b$.
For which positive integers $x,y$ can the stone be moved to $(x,y)$?
The vertices of a triangle are lattice points (they have integer coordinates). There are no other lattice points on the boundary of the triangle, but there is exactly one lattice point inside the triangle. Show that it must be the centroid.
In the beginning, number $1$ has been written to point $(0,0)$ and $0$ has been written to any other point of integral coordinates. After every second, all numbers are replaced with the sum of the numbers in four neighbouring points at the previous second. Find the sum of numbers in all points of integral coordinates after $n$ seconds.
Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.
A cube of integral dimensions is given in space so that all four vertices of one of the faces are lattice points. Prove that the other four vertices are also lattice points.
How many of the following statements are false?
a. $2005$ distinct positive integers exist such that the sum of their squares is a cube and the sum of their cubes is a square.
b. There are $2$ integral solutions to $x^2 + y^2 + z^2 = x^2y^2$.
c. If the vertices of a triangle are lattice points in a plane, the diameter of the triangle’s circumcircle will never exceed the product of the triangle’s side lengths.
Let $A$ be the number of all points with integer coordinates in a three-dimensional coordinate system. We assume that nine arbitrary points in $A$ will be colored blue. Show that we can always find two blue dots so that the line segment between them contains at least one point from $A$.
Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds $1$.
Which is the least positive integer $n$ for which it is possible to find a (non-degenerate) $n$-gon with sidelengths $1, 2,. . . , n$, and where all vertices have integer coordinates?
The plan considers $51$ points of integer coordinates, so that the distances between any two points are natural numbers. Show that at least $49\%$ of the distances are even.
The following geometry is embedded in the Cartesian plane.
[b]a)[/b] Prove that any line that passes through at least two lattice points, passes through at least three lattice points.
[b]b)[/b] Find a point on the plane which doesn't lie on any line that passes through at least two lattice points.
[b]c)[/b] Show that any point with rational coordinates lie on a line that passes through two lattice points.
[i]Lavinia Savu[/i]
It is allowed to perform the following transformations in the plane with any integers $a$:
(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,
(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.
Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:
a) Vertices of a square,
b) Vertices of a rectangle with unequal side lengths?
To each point of $\mathbb{Z}^3$ we assign one of $p$ colors.
Prove that there exists a rectangular parallelepiped with all its vertices in $\mathbb{Z}^3$ and of the same color.
Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.
Given a positive integer $n$, let $M$ be the set of all points in space with integer coordinates $(a, b, c)$ such that $0 \le a, b, c \le n$. A frog must go to the point $(0, 0, 0)$ to the point $(n, n, n)$ according to the following rules:
$\bullet$ The frog can only jump to points of M.
$\bullet$ In each jump, the frog can go from point $(a, b, c)$ to one of the following points: $(a + 1, b, c)$, $(a, b + 1, c)$, $(a, b, c + 1)$, or $(a, b, c - 1)$.
$\bullet$ The frog cannot pass through the same point more than once.
In how many different ways can the frog achieve its goal?
Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is
(a) a triangle with at least two sides longer than $1000$?
(b) a triangle whose sides are all longer than $1000$?
(c) a quadrangle?
Let $ ABC $ be a triangle whose vertices have integer coordinates and inside of which there is exactly one point $ O $ with integer coordinates. Let $ D $ be the intersection of the lines $ BC $ and $ AO. $ Find the largest possible value of $ \frac {\overline{AO}} {\overline{OD}} $.
Let $M$ be the set of all points $(x,y)$ in the cartesian plane, with integer coordinates satisfying $1 \le x \le 12$ and $1 \le y \le 13$.
(a) Prove that every $49$-element subset of $M$ contains four vertices of a rectangle with sides parallel to the coordinate axes.
(b) Give an example of a $48$-element subset of $M$ without this property.
Every vertex of a polygon has both integer coordinates; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number.
Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin.
Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$.
Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?