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.

Let $a\leqslant b\leqslant c$ be non-negative integers. A triangle on a checkered plane with vertices in the nodes of the grid is called an $(a,b,c)$[i]-triangle[/i] if there are exactly $a{}$ nodes on one side of it (not counting vertices), exactly $b{}$ nodes on the second side, and exactly $c{}$ nodes on the third side. [list] [*]Does there exist a $(9,10,11)$-triangle? [*]Find all triples of non-negative integers $a\leqslant b\leqslant c$ for which there exists an $(a,b,c)$-triangle. [*]For each such triple, find the minimum possible area of the $(a,b,c)$-triangle. [/list] [i]Proposed by P. Kozhevnikov[/i]

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

Given a circle centered at the origin. Prove that there is a circle of smaller radius that has no less points with integer coordinates.

Let $n,a,b,c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.

In a plane with a square grid, where the side length of the base square is $1$, lies a right triangle. All its vertices are lattice points and all side lengths are integer. Prove that the center of the incircle is also a lattice point.

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 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?

A parallelogram is constructed on the coordinate plane, the coordinates of which are integers. It is known that inside the parallelogram and on its contour there are other (except vertices) points with integer coordinates. Prove that the area of ​​the parallelogram is not less than $3/2$.

A convex polygon on a plane contains at least $m^2+1$ points with integer coordinates. Prove that it contains $m+1$ points with integer coordinates that lie on the same line.

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.

Let $n$ be a positive integers. Equilateral triangle with sides of length $n$ is split into equilateral triangles with side lengths $1$, forming a triangular lattice. Call an equilateral triangle with vertices in the lattice "important". Let $p_k$ be the number of unordered pairs of vertices in the lattice which participate in exactly $k$ important triangles. Find (as a function of $n$) (a) $p_0+p_1+p_2$ (b) $p_1+2p_2$

In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is $2011$, prove that the sides of the rectangular box are parallel to the coordinate axes.

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 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?

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.

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 $\triangle$.

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 $T$?

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.

Let $ S$ be the set of $25$ points $(x, y)$ with $0\le x, y \le 4$. A triangle whose three vertices are in $S$ is chosen at random. What is the expected value of the square of its area?

A rectangle with the side lengths $a$ and $b$ with $a <b$ should be placed in a right-angled coordinate system so that there is no point with integer coordinates in its interior or on its edge. Under what necessary and at the same time sufficient conditions for $a$ and $b$ is this possible?

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.

Let $P =\{(a, b) / a, b \in \{1, 2, ..., n\}, n \in N\}$ be a set of point of the Cartesian plane and draw horizontal, vertical, or diagonal segments, of length $1$ or $\sqrt 2$, so that both ends of the segment are in $P$ and do not intersect each other. Furthermore, for each point $(a, b)$ it is true that i) if $a + b$ is a multiple of $3$, then it is an endpoint of exactly $3$ segments. ii) if $a + b$ is an even not multiple of $3$, then it is an endpoint of exactly $2$ segments. iii) if $a + b$ is an odd not multiple of $3$, then it is endpoint of exactly $1$ segment. a) Check that with $n = 6$ it is possible to satisfy all the conditions. b) Show that with $n = 2015$ it is not possible to satisfy all the conditions.

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 $20$? $\textbf{(A) }441\qquad\textbf{(B) }761\qquad\textbf{(C) }841\qquad\textbf{(D) }921\qquad\textbf{(E) }924$