Olympiad problems

2013 Junior Balkan Team Selection Tests - Romania, 2

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 $A$

2011 QEDMO 8th, 8

Tags: combinatorial geometry , lattice , coloring , combinatorics

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?

1987 Austrian-Polish Competition, 9

Tags: combinatorial geometry , lattice , parallel , combinatorics

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.

1983 Poland - Second Round, 1

Tags: geometry , combinatorial geometry , lattice , analytic geometry

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.

1992 Poland - Second Round, 1

Tags: analytic geometry , lattice , combinatorial geometry , geometry

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.

1984 Poland - Second Round, 5

Tags: geometry , lattice , geometric inequalities , analytic geometry

Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.

1967 Swedish Mathematical Competition, 6

Tags: combinatorial geometry , lattice , triangle area , geometry

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).

2013 Saudi Arabia BMO TST, 1

Tags: lattice , combinatorial geometry , combinatorics

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 $|x_1 - x_2| = |x_2 - x_3| = |x_3 - x_4|=|x_4 - x_5| = 1$.

2004 Estonia National Olympiad, 4

Tags: combinatorics , lattice , sum

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.

2005 Abels Math Contest (Norwegian MO), 2b

Tags: coloring , lattice , combinatorics

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 $A$.

2019 Estonia Team Selection Test, 6

Tags: geometry , lattice , rhombus , combinatorics

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?

1994 North Macedonia National Olympiad, 2

Tags: lattice , ratio , geometry

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}} $.

2017 Thailand Mathematical Olympiad, 10

Tags: combinatorial geometry , geometry , lattice points , lattice , interior , circles

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).

1982 Kurschak Competition, 1

Tags: combinatorial geometry , 3d geometry , lattice , geometry

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.

1978 Dutch Mathematical Olympiad, 4

Tags: combinatorics , combinatorial geometry , lattice

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.

2024 Bulgarian Autumn Math Competition, 9.3,9.4

Tags: number theory , lattice , squarefree

$9.3$ 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 $300$.

1997 ITAMO, 3

Tags: combinatorial geometry , combinatorics , lattice , square lattice , coloring

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?

1998 Singapore MO Open, 2

Tags: inequalities , combinatorics , function , line , lattice , collinear , combinatorial geometry

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 $(n,f(n))$.

2024 IMAR Test, P4

Tags: analytic geometry , geometry , combinatorics , lattice

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.

2023 China Team Selection Test, P12

Tags: combinatorial geometry , combinatorics , analytic geometry , lattice

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; [*] $AB, BC, CA>D$; [*] $S_{\triangle ABC}<\frac{\sqrt 3}{4}D^2+\lambda\cdot D^{4/5}$.

2020 Spain Mathematical Olympiad, 3

Tags: combinatorics , lattice

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.

1998 Moldova Team Selection Test, 11

Tags: geometry , ratio , lattice points , lattice , maximum

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.