Found problems: 99
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?
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 $f(r) = f(s)$.
2024 Bulgarian Autumn Math Competition, 8.4
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) $p_1+2p_2$
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).
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?
2023 Olympic Revenge, 4
Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?
1994 Tuymaada Olympiad, 8
Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.
2018 China National Olympiad, 2
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.
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 $3\sqrt5$.
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.
2019 China Girls Math Olympiad, 4
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 $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.
1982 Brazil National Olympiad, 3
$S$ is a $(k+1) \times (k+1)$ array of lattice points. How many squares have their vertices in $S$?
2024 Canadian Mathematical Olympiad Qualification, 5
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?
1978 Germany Team Selection Test, 6
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.
2024 New Zealand MO, 7
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?
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.
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)
1994 Nordic, 2
We call a finite plane set $S$ consisting of points with integer coefficients a two-neighbour set, if for each point $(p, q)$ of $S$ exactly two of the points $(p +1, q), (p, q +1), (p-1, q), (p, q-1)$ belong to $S$. For which integers $n$ there exists a two-neighbour set which contains exactly $n$ points?
1987 All Soviet Union Mathematical Olympiad, 457
Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points.
2014 Contests, 3
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.
1991 Bundeswettbewerb Mathematik, 3
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.
1990 Mexico National Olympiad, 5
Given $19$ points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates.
2022 Romania Team Selection Test, 5
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 $S.$
Kvant 2020, M2619
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]
2021 Dutch Mathematical Olympiad, 3
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.