Consider the curve $\Gamma$ defined by the equation $y^2 = x^3 +bx+b^2$, where $b$ is a nonzero rational constant. Inscribe in the curve $\Gamma$ a triangle whose vertices have rational coordinates.

Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is its area? (A)$20\sqrt{3}$ (B)$20\sqrt{2}$ (C)$25\sqrt{3}$ (D)None of these

An ant is moving on the cooridnate plane, starting form point $(0,-1)$ along a straight line until it reaches the $x$- axis at point $(x,0)$ where $x$ is a real number. After it turns $90^o$ to the left and moves again along a straight line until it reaches the $y$-axis . Then it again turns left and moves along a straight line until it reaches the $x$-axis, where it once more turns left by $90^o$ and moves along a straight line until it finally reached the $y$-axis. Can both the length of the ant's journey and distance between it's initial and final point be: (a) rational numbers ? (b) integers? Justify your answers PS. Collected [url=https://artofproblemsolving.com/community/c949609_2019_nigerian_senior_mo_round_4]here[/url]

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Describe the amount of points $P(x, y)$ that are twice as far apart $A(3, 0)$ as to $0(0, 0)$.

Let $y = x^2 + ax + b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$.

Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.

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.

A grasshopper is hopping in the angle $x\ge 0, y\ge 0$ of the coordinate plane (that means that it cannot land in the point with negative coordinate). If it is in the point $(x,y)$, it can either jump to the point $(x+1,y-1)$, or to the point $(x-5,y+7)$. Draw a set of such an initial points $(x,y)$, that having started from there, a grasshopper cannot reach any point farther than $1000$ from the point $(0,0)$. Find its area.

The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point? $\textbf{(A)}\ 4\sqrt2 \qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 5\sqrt2 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 6\sqrt2$

Consider the points of intersection of the graphs $y = \cos x$ and $x = 100 \cos (100y)$ for which both coordinates are positive. Let $a$ be the sum of their $x$-coordinates and $b$ be the sum of their $y$-coordinates. Determine the value of $\frac{a}{b}$.

The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$. What is $c$? $\textbf{(A) } -13\qquad \textbf{(B) } -8\qquad \textbf{(C) } 2\qquad \textbf{(D) } 8\qquad \textbf{(E) } 13$

For all integers $x,y$, a non-negative integer $f(x,y)$ is written on the point $(x,y)$ on the coordinate plane. Initially, $f(0,0) = 4$ and the value written on all remaining points is $0$. For integers $n, m$ that satisfies $f(n,m) \ge 2$, define '[color=#9a00ff]Seehang[/color]' as the act of reducing $f(n,m)$ by $1$, selecting 3 of $f(n,m+1), f(n,m-1), f(n+1,m), f(n-1,m)$ and increasing them by 1. Prove that after a finite number of '[color=#0f0][color=#9a00ff]Seehang[/color][/color]'s, it cannot be $f(n,m)\le 1$ for all integers $n,m$.

Is it true that for each natural number $n$ there exist a circle, which contains exactly $n$ points with integer coordinates?

The graphs of four functions of the form $y = x^2 + ax + b$, where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest $x$-coordinates of the points of intersection is equal to the sum of the other two. [i](3 points)[/i]

A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.

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.

The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 13$

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point $(a, b)$ is one greater than the average of the numbers written on points $ (a+1 , b-1) $ and $ (a-1,b+1)$ . Which numbers could he write on point $(121, 212)$? Note: The elements of the first quadrant are points where both of the coordinates are non- negative.

Let the [i]taxicab distance[/i] between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane is given by $|x_1-x_2|+|y_1-y_2|$. For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$? $\textbf{(A) }441\qquad\textbf{(B) }761\qquad\textbf{(C) }841\qquad\textbf{(D) }921\qquad\textbf{(E) }924$

A moving particle starts at the point $\left(4,4\right)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $\left(a,b\right)$, it moves at random to one of the points $\left(a-1,b\right)$, $\left(a,b-1\right)$, or $\left(a-1,b-1\right)$, each with probability $\tfrac{1}{3}$, independently of its previous moves. The probability that it will hit the coordinate axes at $\left(0,0\right)$ is $\tfrac{m}{3^n}$, where $m$ and $n$ are positive integers, and $m$ is not divisible by $3$. Find $m+n$.

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Through the point $F(0,\frac{1}{4})$ of the coordinate plane two perpendicular lines pass, that intersect parabola $y=x^2$ at points $A,B,C,D$ ($A_x<B_x<C_x<D_x$) The difference of projections of segments $AD$ and $BC$ onto the $Ox$ line is $m$ Find the area of $ABCD$