A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

Define a [b]ring[/b] in the plane to be the set of points at a distance of at least $r$ and at most $R$ from a specific point $O$, where $r<R$ are positive real numbers. Rings are determined by the three parameters $(O, R, r)$. The area of a ring is labeled $S$. A point in the plane for which both its coordinates are integers is called an integer point. [b]a)[/b] For each positive integer $n$, show that there exists a ring not containing any integer point, for which $S>3n$ and $R<2^{2^n}$. [b]b)[/b] Show that each ring satisfying $100\cdot R<S^2$ contains an integer point.

The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.

Consider the set of all integer points in $Z^3$. Sasha and Masha play such a game. At first, Masha marks an arbitrary point. After that, Sasha marks all the points on some a plane perpendicular to one of the coordinate axes and at no point, which Masha noted. Next, they continue to take turns (Masha can't to select previously marked points, Sasha cannot choose the planes on which there are points said Masha). Masha wants to mark $n$ consecutive points on some line that parallel to one of the coordinate axes, and Sasha seeks to interfere with it. Find all $n$, in which Masha can achieve the desired result.

A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.

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.

There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points with exactly $1$ unit apart. Find the maximum possible value of $I$. ([i]Note. An integer point is a point with integer coordinates.[/i]) [i]Proposed by CSJL.[/i]

Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?

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?

Is it possible to write natural numbers at all points of the plane with integer coordinates so that three points with integer coordinates lie on the same line if and only if the numbers written in them had a common divisor greater than one?

Let $m$ and $n$ be positive integers. Let $S$ be the set of all points $(x, y)$ with integer coordinates such that $1 \leq x,y \leq m+n-1$ and $m+1 \leq x +y \leq 2m+n-1.$ Let $L$ be the set of the $3m+3n-3$ lines parallel to one of $x = 0, y = 0,$ or $x + y = 0$ and passing through at least one point in $S$. For which pairs $(m, n)$ does there exist a subset $T$ of $S$ such that every line in $L$ intersects an odd number of elements of $T$?

Some lattice points in the Cartesian coordinate system are colored red, the rest of the lattice points are colored blue. Such a coloring is called [i]finitely universal[/i], if for any finite, non-empty $A\subset \mathbb Z$ there exists $k\in\mathbb Z$ such that the point $(x,k)$ is colored red if and only if $x\in A$. $a)$ Does there exist a finitely universal coloring such that each row has finitely many lattice points colored red, each row is colored differently, and the set of lattice points colored red is connected? $b)$ Does there exist a finitely universal coloring such that each row has a finite number of lattice points colored red, and both the set of lattice points colored red and the set of lattice points colored blue are connected? A set $H$ of lattice points is called [i]connected[/i] if, for any $x,y\in H$, there exists a path along the grid lines that passes only through lattice points in $H$ and connects $x$ to $y$. [i]Submitted by Anett Kocsis, Budapest[/i]

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 $(x,y)$?

A point in the plane is called [b]integral[/b] if both its $x$ and $y$ coordinates are integers. We are given a triangle whose vertices are integral. Its sides do not contain any other integral points. Inside the triangle, there are exactly 4 integral points. Must those 4 points lie on one line?

The lattice point on the plane is a point that has coordinates in the form of a pair of integers. Let $P_1, P_2, P_3, P_4, P_5$ be five different lattice points on the plane. Prove that there is a pair of points $(P_i, P_j), i \ne j$, so that the line segment $P_iP_j$ contains a lattice point other than $P_i$ and $P_j$.

The coordinates of the triangle's vertices in the Cartesian system $XOY$ are integers. Prove that the diameter of the circle circumscribed by this triangle is not greater than the product of the lengths of the triangle's sides.

Fredy starts at the origin of the Euclidean plane. Each minute, Fredy may jump a positive integer distance to another lattice point, provided the jump is not parallel to either axis. Can Fredy reach any given lattice point in 2023 jumps or less? [i]Proposed by Tony Wang[/i]

The Fibonacci sequence is given by equalities $$F_1=F_2=1, F_{k+2}=F_k+F_{k+1}, k\in N$$. a) Prove that for every $m \ge 0$, the area of ​​the triangle $A_1A_2A_3$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$ is equal to $0.5$. b) Prove that for every $m \ge 0$ the quadrangle $A_1A_2A_4$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$, $A_4 (F_{m+7},F_{m+8})$ is a trapezoid, whose area is equal to $2.5$. c) Prove that the area of ​​the polygon $A_1A_2...A_n$ , $n \ge3$ with vertices does not depend on the choice of numbers $m \ge 0$, and find this area.

$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

Prove that there are infinitely many different triangles in coordinate plane satisfying: 1) their vertices are lattice points 2) their side lengths are consecutive integers [b]Remark[/b]: Triangles that can be obtained by rotation or translation or shifting the coordinate system are considered as equal triangles

Let the [i]taxicab distance[/i] between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane is given by $|x_1-x_2|+|y_1-y_2|$. For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$? $\textbf{(A) }441\qquad\textbf{(B) }761\qquad\textbf{(C) }841\qquad\textbf{(D) }921\qquad\textbf{(E) }924$

Consider the part of a lattice given by the corners $(0, 0), (n, 0), (n, 2)$ and $(0, 2)$. From a lattice point $(a, b)$ one can move to $(a + 1, b)$ or to $(a + 1, b + 1)$ or to $(a, b - 1$), provided that the second point is also contained in the part of the lattice. How many ways are there to move from $(0, 0)$ to $(n, 2)$ considering these rules?

In the coordinate plane, consider the set of all segments of integer lengths whose endpoints have integer coordinates. Prove that no two of these segments form an angle of $45^o$. Are there such segments in coordinate space?

In Cartesian space, let $\Omega = \{(a, b, c) : a, b, c$ are integers between $1$ and $30\}$. A point of $\Omega$ is said to be [i]visible [/i] from the origin if the segment that joins said point with the origin does not contain any other elements of $\Omega$. Find the number of points of $\Omega$ that are [i]visible [/i] from the origin.

Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.