2016 AIME I Problems/Problem 14

Problem

Centered at each lattice point in the coordinate plane are a circle radius $\frac{1}{10}$ and a square with sides of length $\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$ .

Solution

First note that $1001 = 143 \cdot 7$ and $429 = 143 \cdot 3$ so every point of the form $(7k, 3k)$ is on the line. Then consider the line $l$ from $(7k, 3k)$ to $(7(k + 1), 3(k + 1))$ . Translate the line $l$ so that $(7k, 3k)$ is now the origin. There is one square and one circle that intersect the line around $(0,0)$ . Then the points on $l$ with an integral $x$ -coordinate are, since $l$ has the equation $y = \frac{3x}{7}$ :

$(0,0), (1, \frac{3}{7}), (2, \frac{6}{7}), (3, 1 + \frac{2}{7}), (4, 1 + \frac{5}{7}), (5, 2 + \frac{1}{7}), (6, 2 + \frac{4}{7}), (7,3).$

We claim that the lower right vertex of the square centered at $(2,1)$ lies on $l$ . Since the square has side length $\frac{1}{5}$ , the lower right vertex of this square has coordinates $(2 + \frac{1}{10})$ ,

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2016 AIME I Problems/Problem 14

Problem

Solution