Found problems: 47
1998 Moldova Team Selection Test, 11
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.
1988 China Team Selection Test, 3
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.
2005 iTest, 30
How many of the following statements are false?
a. $2005$ distinct positive integers exist such that the sum of their squares is a cube and the sum of their cubes is a square.
b. There are $2$ integral solutions to $x^2 + y^2 + z^2 = x^2y^2$.
c. If the vertices of a triangle are lattice points in a plane, the diameter of the triangle’s circumcircle will never exceed the product of the triangle’s side lengths.
2004 Nicolae Păun, 2
The following geometry is embedded in the Cartesian plane.
[b]a)[/b] Prove that any line that passes through at least two lattice points, passes through at least three lattice points.
[b]b)[/b] Find a point on the plane which doesn't lie on any line that passes through at least two lattice points.
[b]c)[/b] Show that any point with rational coordinates lie on a line that passes through two lattice points.
[i]Lavinia Savu[/i]
2024 Bulgarian Autumn Math Competition, 9.3,9.4
$9.3$
A natural number is called square-free, if it is not divisible by the square of any prime number. For a natural number $a$, we consider the number $f(a) = a^{a+1} + 1$. Prove that:
a) if $a$ is even, then $f(a)$ is not square-free
b) there exist infinitely many odd $a$ for which $f(a)$ is not square-free
$9.4$
We will call a generalized $2n$-parallelogram a convex polygon with $2n$ sides, so that, traversed consecutively, the $k$th side is parallel and equal to the $(n+k)$th side for $k=1, 2, ... , n$. In a rectangular coordinate system, a generalized parallelogram is given with $50$ vertices, each with integer coordinates. Prove that its area is at least $300$.
2023 China Team Selection Test, P12
Prove that there exists some positive real number $\lambda$ such that for any $D_{>1}\in\mathbb{R}$, one can always find an acute triangle $\triangle ABC$ in the Cartesian plane such that [list] [*] $A, B, C$ lie on lattice points; [*] $AB, BC, CA>D$; [*] $S_{\triangle ABC}<\frac{\sqrt 3}{4}D^2+\lambda\cdot D^{4/5}$.
1966 Swedish Mathematical Competition, 5
Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin.
Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$.
Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?
2012 Bundeswettbewerb Mathematik, 4
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?
2019 Estonia Team Selection Test, 6
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?
1997 ITAMO, 3
The positive quadrant of a coordinate plane is divided into unit squares by lattice lines. Is it possible to color the squares in black and white so that:
(i) In every square of side $n$ ($n \in N$) with a vertex at the origin and sides are parallel to the axes, there are more black than white squares;
(ii) Every diagonal parallel to the line $y = x$ intersects only finitely many black squares?
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).
2012 QEDMO 11th, 5
Let $n$ be a natural number and $L = Z^2$ the set of points on the plane with integer coordinates. Every point in $L$ is colored now in one of the colors red or green. Show that there are $n$ different points $x_1,...,x_n \in L$ all of which have the same color and whose center of gravity is also in $L$ and is of the same color.
1983 Poland - Second Round, 1
On a plane with a fixed coordinate system, there is a convex polygon whose all vertices have integer coordinates. Prove that twice the area of this polygon is an integer.
1978 Dutch Mathematical Olympiad, 4
On the plane with a rectangular coordinate system, a set of infinitely many rectangles is given. Every rectangle has the origin as one of its vertices. The sides of all rectangles are parallel to the coordinate axes, and all sides have integer lengths. Prove that there are at least two rectangles in the set, one of which completely covers the other.
1941 Eotvos Mathematical Competition, 2
Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds $1$.
1992 Poland - Second Round, 1
Every vertex of a polygon has both integer coordinates; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number.
1973 Kurschak Competition, 2
For any positive real $r$, let $d(r)$ be the distance of the nearest lattice point from the circle center the origin and radius $r$. Show that $d(r)$ tends to zero as $r$ tends to infinity.
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.
1984 Poland - Second Round, 5
Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.
1994 North Macedonia National Olympiad, 2
Let $ ABC $ be a triangle whose vertices have integer coordinates and inside of which there is exactly one point $ O $ with integer coordinates. Let $ D $ be the intersection of the lines $ BC $ and $ AO. $ Find the largest possible value of $ \frac {\overline{AO}} {\overline{OD}} $.
2013 Saudi Arabia IMO TST, 1
Adel draws an $m \times n$ grid of dots on the coordinate plane, at the points of integer coordinates $(a,b)$ where $1 \le a \le m$ and $1 \le b \le n$. He proceeds to draw a closed path along $k$ of these dots, $(a_1, b_1)$,$(a_2,b_2)$,...,$(a_k,b_k)$, such that $(a_i,b_i)$ and $(a_{i+1}, b_{i+1})$ (where $(a_{k+1}, b_{k+1}) = (a_1, b_1)$) are $1$ unit apart for each $1 \le i \le k$. Adel makes sure his path does not cross itself, that is, the $k$ dots are distinct. Find, with proof, the maximum possible value of $k$ in terms of $m$ and $n$.
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?