Found problems: 99
2022 USAMTS Problems, 5
A lattice point is a point on the coordinate plane with integer coefficients. Prove or disprove : there exists a finite set $S$ of lattice points such that for every line $l$ in the plane with slope $0,1,-1$, or undefined, either $l$ and $S$ intersect at exactly $2022$ points, or they do not intersect.
2017 Thailand Mathematical Olympiad, 10
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).
Russian TST 2015, P1
Let $P(x, y)$ and $Q(x, y)$ be polynomials in two variables with integer coefficients. The sequences of integers $a_0, a_1,\ldots$ and $b_0, b_1,\ldots$ satisfy \[a_{n+1}=P(a_n,b_n),\quad b_{n+1}=Q(a_n,b_n)\]for all $n\geqslant 0$. Let $m_n$ be the number of integer points of the coordinate plane, lying strictly inside the segment with endpoints $(a_n,b_n)$ and $(a_{n+1},b_{n+1})$. Prove that the sequence $m_0,m_1,\ldots$ is non-decreasing.
Russian TST 2021, P2
The natural numbers $t{}$ and $q{}$ are given. For an integer $s{}$, we denote by $f(s)$ the number of lattice points lying in the triangle with vertices $(0;-t/q), (0; t/q)$ and $(t; ts/q)$. Suppose that $q{}$ divides $rs-1{}$. Prove that $f(r) = f(s)$.
1972 Poland - Second Round, 3
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.
1976 Bundeswettbewerb Mathematik, 1
Nine lattice points (i.e. with integer coordinates) $P_1,P_2,...,P_9$ are given in space. Show that the midpoint of at least one of the segments $P_iP_j$ , where $1 \le i < j \le 9$, is a lattice point as well.
Durer Math Competition CD 1st Round - geometry, 2009.D4
If all vertices of a triangle on the square grid are grid points, then the triangle is called a [i]lattice[/i] triangle. What is the area of the lattice triangle with (one) of the smallest area, if one grid has area $1$ square unit?
1982 Brazil National Olympiad, 3
$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$?
1969 IMO Shortlist, 20
$(FRA 3)$ 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 $T = 2S- B -2I + 2.$
2022 Taiwan TST Round 2, C
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]
1977 IMO Longlists, 5
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.
KoMaL A Problems 2021/2022, A. 823
For positive integers $n$ consider the lattice points $S_n=\{(x,y,z):1\le x\le n, 1\le y\le n, 1\le z\le n, x,y,z\in \mathbb N\}.$ Is it possible to find a positive integer $n$ for which it is possible to choose more than $n\sqrt{n}$ lattice points from $S_n$ such that for any two chosen lattice points at least two of the coordinates of one is strictly greater than the corresponding coordinates of the other?
[I]Proposed by Endre Csóka, Budapest[/i]
1982 Czech and Slovak Olympiad III A, 3
In the plane with coordinates $x,y$, find an example of a convex set $M$ that contains infinitely many lattice points (i.e. points with integer coordinates), but at the same time only finitely many lattice points from $M$ lie on each line in that plane.
Indonesia Regional MO OSP SMA - geometry, 2004.5
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 $P_j$.
1994 Tuymaada Olympiad, 8
Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.
2020 June Advanced Contest, 3
Let a [i]lattice tetrahedron[/i] denote a tetrahedron whose vertices have integer coordinates. Given a lattice tetrahedron, a [i]move[/i] consists of picking some vertex and moving it parallel to one of the three edges of the face opposite the vertex so that it lands on a different point with integer coordinates. Prove that any two lattice tetrahedra with the same volume can be transformed into each other by a series of moves
2009 Stars Of Mathematics, 3
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.
2009 Abels Math Contest (Norwegian MO) Final, 3b
Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)
1986 All Soviet Union Mathematical Olympiad, 422
Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.
1991 Spain Mathematical Olympiad, 1
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?
2020 Peru IMO TST, 3
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?
2022 Mexican Girls' Contest, 3
Consider a set $S$ of $16$ lattice points. The $16$ points of $S$ are divided into $8$ pairs in such a way that
[i]for every point $A$ and any of the $7$ pairs of points $(B,C)$ where $A$ is not included, $A$ is at a distance of at most $\sqrt{5}$ from either $B$ or $C$[/i]
Prove that any two points in the set $S$ are at a distance of at most $3\sqrt5$.
1995 Tournament Of Towns, (459) 4
Some points with integer coordinates in the plane are marked. It is known that no four of them lie on a circle. Show that there exists a circle of radius 1995 without any marked points inside.
(AV Shapovelov)
2023 Chile National Olympiad, 2
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.