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.

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 $A$

Albatross and Frankinfueter are playing again: each of them takes turns choosing one point in the plane with integer coordinates and paint it in his favorite color. Albatross plays first. Someone wins as soon as there is a square with all four corners in the are colored in their own color. Does anyone has a winning strategy and if so, who?

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.

Consider all trapezoids in a coordinate plane with interior angles of $90^o, 90^o, 45^o$ and $135^o$ whose bases are parallel to a coordinate axis and whose vertices have integer coordinates. Define the [i]size [/i] of such a trapezoid as the total number of points with integer coordinates inside and on the boundary of the trapezoid. (a) How many pairwise non-congruent such trapezoids of size $2001$ are there? (b) Find all positive integers not greater than $50$ that do not appear as sizes of any such trapezoid.

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.

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.

A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold: (i) The endpoints of each selected segment are lattice points; (ii) Each selected segment is parallel to a coordinate axis or to one of the lines $y = \pm x$, (iii) Each selected segment contains exactly five lattice points, all of which are selected, (iv) Every two selected segments have at most one common point. A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.

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.

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?

In the Cartesian plane is given a polygon $P$ whose vertices have integer coordinates and with sides parallel to the coordinate axes. Show that if the length of each edge of $P$ is an odd integer, then the surface of P cannot be partitioned into $2\times 1$ rectangles.

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 vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

The set $G$ is defined by the points $(x,y)$ with integer coordinates, $1 \le x \le 5$ and $1 \le y \le 5$. Determine the number of five-point sequences $(P_1, P_2, P_3, P_4, P_5)$ such that for $1 \le i \le 5$, $P_i = (x_i,i)$ is in $G$ and $|x_1 - x_2| = |x_2 - x_3| = |x_3 - x_4|=|x_4 - x_5| = 1$.

For a given positive integer $n$, let $A_n$ be the set of all points $(x,y)$ in the coordinate plane with $x,y \in \{0,1,...,n\}$. A point $(i, j)$ is called internal if $0 < i, j < n$. A real function $f$ , defined on $A_n$, is called [i]good [/i] if it has the following property: For every internal point $x$, the value of $f(x)$ is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance $1$ from $x$). Prove that if $f$ and $g$ are good functions that coincide at the non-internal points of $A_n$, then $f \equiv g$.

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.

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.

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 $A$.

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 $(x,y)$?

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 $M$ and $N$ be positive integers. Pisut walks from point $(0, N)$ to point $(M, 0)$ in steps so that $\bullet$ each step has unit length and is parallel to either the horizontal or the vertical axis, and $\bullet$ each point ($x, y)$ on the path has nonnegative coordinates, i.e. $x, y > 0$. During each step, Pisut measures his distance from the axis parallel to the direction of his step, if after the step he ends up closer from the origin (compared to before the step) he records the distance as a positive number, else he records it as a negative number. Prove that, after Pisut completes his walk, the sum of the signed distances Pisut measured is zero.

A [i]diameter[/i] of a finite planar set is any line segment of maximal Euclidean length having both end points in that set. A [i]lattice point[/i] in the Cartesian plane is one whose coordinates are both integral. Given an integer $n\geq 2$, prove that a set of $n$ lattice points in the plane has at most $n-1$ diameters.

The vertices of a triangle are lattice points. There are no lattice points on the sides (apart from the vertices) and $n$ lattice points inside the triangle. Show that its area is $n + \frac12$. Find the formula for the general case where there are also $m$ lattice points on the sides (apart from the vertices).

Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.