Given parallelogram $OABC$ in the coodinate with $O$ the origin and $A,B,C$ be lattice points. Prove that for all lattice point $P$ in the internal or boundary of $\triangle ABC$, there exists lattice points $Q,R$(can be the same) in the internal or boundary of $\triangle OAC$ with $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.

$(FRA 3)$ 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 $T = 2S- B -2I + 2.$

Show that for each integer $n > 0$, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into $1 \times 2$ (and / or $2 \times 1$) rectangles in exactly $n$ ways.

The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than $4$.

All points in the plane that have both integer coordinates are painted, using the colors red, green, and yellow. If the points are painted so that there is at least one point of each color. Prove that there are always three points $X$, $Y$ and $Z$ of different colors, such that $\angle XYZ = 45^{\circ} $

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

Let $n$ and $k$ be positive integers and let $$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$ be the length $n$ lattice cube. Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red such that if $P$ and $Q$ are red points and $PQ$ is parallel to one of the coordinate axes, then the whole line segment $PQ$ consists of only red points. Prove that there exists at least $k$ unit cubes of length $1$, all of whose vertices are colored red.

[b](9.7)[/b] On the segment $[0, 2002]$ its ends and the point with coordinate $d$ are marked, where $d$ is a coprime number to $1001$. It is allowed to mark the midpoint of any segment with ends at the marked points, if its coordinate is integer. Is it possible, by repeating this operation several times, to mark all the integer points on a segment? [b](10.7)[/b] On the segment $[0, 2002]$ its ends and $n-1 > 0$ integer points are marked so that the lengths of the segments into which the segment $ [0, 2002]$ is divided are corpime in the total (i.e., have no common divisor greater than $1$). It is allowed to divide any segment with marked ends into $n$ equal parts and mark the division points if they are all integers. (The point can be marked a second time, but it remains marked.) Is it possible, by repeating this operation several times, mark all the integer points on the segment? [b](11.8)[/b] On the segment $ [0,N]$ its ends and $2 $ more points are marked so that the lengths segments into which the segment $[0,N]$ is divided are integer and coprime in total. If there are two marked points $A$ and $B$ such that the distance between them is a multiple of $3$, then we can divide from cutting $AB$ by $3$ equal parts, mark one of the division points and erase one of the points $A, B$. Is it true that for several such actions you can mark any predetermined integer point of the segment $[0,N]$?

Consider the line (E): $5x-10y+3=0$ . Prove that: a) Line $(E)$ doesn't pass through points with integer coordinates. b) There is no point $A(a_1,a_2)$ with $ a_1,a_2 \in \mathbb{Z}$ with distance from $(E)$ less then $\frac{\sqrt3}{20}$.

Let $m,n\geq 2$ be positive integers and $S\subseteq [1,m]\times [1,n]$ be a set of lattice points. Prove that if \[|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloor\]then there exists a circle which passes through at least four distinct points of $S.$

[b]4.[/b] We see, in Figures 1 and 2, examples of lock screens from a cellphone that only works with a password that is not typed but drawn with straight line segments. Those segments form a polygonal line with vertices in a lattice. When drawing the pattern that corresponds to a password, the finger can't lose contact with the screen. Every polygonal line corresponds to a sequence of digits and this sequence is, in fact, the password. The tracing of the polygonal obeys the following rules: [i]i.[/i] The tracing starts at some of the detached points which correspond to the digits from $1$ to $9$ (Figure 3). [i]ii.[/i] Each segment of the pattern must have as one of its extremes (on which we end the tracing of the segment) a point that has not been used yet. [i]iii.[/i] If a segment connects two points and contains a third one (its middle point), then the corresponding digit to this third point is included in the password. That does not happen if this point/digit has already been used. [i]iv.[/i] Every password has at least four digits. Thus, every polygonal line is associated to a sequence of four or more digits, which appear in the password in the same order that they are visited. In Figure 1, for instance, the password is 218369, if the first point visited was $2$. Notice how the segment connecting the points associated with $3$ and $9$ includes the points associated to digit $6$. If the first visited point were the $9$, then the password would be $963812$. If the first visited point were the $6$, then the password would be $693812$. In this case, the $6$ would be skipped, because it can't be repeated. On the other side, the polygonal line of Figure 2 is associated to a unique password. Determine the smallest $n (n \geq 4)$ such that, given any subset of $n$ digits from $1$ to $9$, it's possible to elaborate a password that involves exactly those digits in some order.

Given a simple number $p>3$. Consider the set $M$ of the pairs $(x,y)$ with the integer coordinates in the plane such that $0 \le x < p, 0 \le y < p$. Prove that it is possible to mark $p$ points of $M$ such that not a triple of marked points will belong to one line and there will be no parallelogram with the vertices in the marked points.

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]

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]

Given a prime $p$ and a real number $\lambda \in (0,1)$. Let $s$ and $t$ be positive integers such that $s \leqslant t < \frac{\lambda p}{12}$. $S$ and $T$ are sets of $s$ and $t$ consecutive positive integers respectively, which satisfy $$\left| \left\{ (x,y) \in S \times T : kx \equiv y \pmod p \right\}\right| \geqslant 1 + \lambda s.$$Prove that there exists integers $a$ and $b$ that $1 \leqslant a \leqslant \frac{1}{ \lambda}$, $\left| b \right| \leqslant \frac{t}{\lambda s}$ and $ka \equiv b \pmod p$.

Call a point in the Cartesian plane with integer coordinates a $lattice$ $point$. Given a finite set $\mathcal{S}$ of lattice points we repeatedly perform the following operation: given two distinct lattice points $A, B$ in $\mathcal{S}$ and two distinct lattice points $C, D$ not in $\mathcal{S}$ such that $ACBD$ is a parallelogram with $AB > CD$, we replace $A, B$ by $C, D$. Show that only finitely many such operations can be performed. [I]Proposed by Joe Benton, United Kingdom.[/i]

In a rectangle with vertices $(0, 0), (0, 2), (n,0),(n, 2)$, ($n$ is a positive integer) find the number of longest paths starting from $(0, 0)$ and arriving at $(n, 2)$ which satisfy the following: $\bullet$ At each movement, you can move right, up, left, down by $1$. $\bullet$ You cannot visit a point you visited before. $\bullet$ You cannot move outside the rectangle.

A lattice point is a point $(x, y)$ with both $x$ and $y$ integers. Find, with proof, the smallest $n$ such that every set of $n$ lattice points contains three points that are the vertices of a triangle with integer area. (The triangle may be degenerate, in other words, the three points may lie on a straight line and hence form a triangle with area zero.)

A frog jumps around on the grid points in the plane, from one grid point to another. The frog starts at the point $(0, 0)$. Then it makes, successively, a jump of one step horizontally, a jump of $2$ steps vertically, a jump of $3$ steps horizontally, a jump of $4$ steps vertically, et cetera. Determine all $n > 0$ such that the frog can be back in $(0, 0)$ after $n$ jumps.

In the plane, construct as many lines in general position as possible, with any two of them intersecting in a point with integer coordinates.

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.

Some of the $80960$ lattice points in a $40\times2024$ lattice are coloured red. It is known that no four red lattice points are vertices of a rectangle with sides parallel to the axes of the lattice. What is the maximum possible number of red points in the lattice?

Find all positive integer $k$ such that one can find a number of triangles in the Cartesian plane, the centroid of each triangle is a lattice point, the union of these triangles is a square of side length $k$ (the sides of the square are not necessarily parallel to the axis, the vertices of the square are not necessarily lattice points), and the intersection of any two triangles is an empty-set, a common point or a common edge.

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.

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.