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Difference between revisions of "2016 AIME I Problems/Problem 14"

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Centered at each lattice point in the coordinate plane are a circle radius <math>\frac{1}{10}</math> and a square with sides of length <math>\frac{1}{5}</math> whose sides are parallel to the coordinate axes. The line segment from <math>(0,0)</math> to <math>(1001, 429)</math> intersects <math>m</math> of the squares and <math>n</math> of the circles. Find <math>m + n</math>.
 
Centered at each lattice point in the coordinate plane are a circle radius <math>\frac{1}{10}</math> and a square with sides of length <math>\frac{1}{5}</math> whose sides are parallel to the coordinate axes. The line segment from <math>(0,0)</math> to <math>(1001, 429)</math> intersects <math>m</math> of the squares and <math>n</math> of the circles. Find <math>m + n</math>.
  
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== Solution ==
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== Solution 1 ==
  
 
First note that <math>1001 = 143 \cdot 7</math> and <math>429 = 143 \cdot 3</math> so every point of the form <math>(7k, 3k)</math> is on the line.  Then consider the line <math>l</math> from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>.  Translate the line <math>l</math> so that <math>(7k, 3k)</math> is now the origin.  There is one square and one circle that intersect the line around <math>(0,0)</math>.  Then the points on <math>l</math> with an integral <math>x</math>-coordinate are, since <math>l</math> has the equation <math>y = \frac{3x}{7}</math>:
 
First note that <math>1001 = 143 \cdot 7</math> and <math>429 = 143 \cdot 3</math> so every point of the form <math>(7k, 3k)</math> is on the line.  Then consider the line <math>l</math> from <math>(7k, 3k)</math> to <math>(7(k + 1), 3(k + 1))</math>.  Translate the line <math>l</math> so that <math>(7k, 3k)</math> is now the origin.  There is one square and one circle that intersect the line around <math>(0,0)</math>.  Then the points on <math>l</math> with an integral <math>x</math>-coordinate are, since <math>l</math> has the equation <math>y = \frac{3x}{7}</math>:
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for(int i=0;i<8;++i)for(int j=0;j<4;++j){dot((i,j),linewidth(1));draw(box((i-.1,j-.1),(i+.1,j+.1)),linewidth(.5));draw(circle((i,j),.1),linewidth(.5));}
 
for(int i=0;i<8;++i)for(int j=0;j<4;++j){dot((i,j),linewidth(1));draw(box((i-.1,j-.1),(i+.1,j+.1)),linewidth(.5));draw(circle((i,j),.1),linewidth(.5));}
 
</asy>
 
</asy>
 +
 +
==Solution 2==
 +
 +
This is mostly a clarification since Solution 1 reads a little confusingly near the end, but let's take the diagram for the origin to <math>(7,3)</math>. We have the origin circle and square intersected, then two squares, then the circle and square at <math>(7,3)</math>. If we take the circle and square at the origin out of the diagram, we will be able to repeat the resulting segment (with its circles and squares) end to end from <math>(0,0)</math> to <math>(1001,429)</math>, which forms the line we need without overlapping. Since <math>143</math> of these segments are needed to do this, and <math>3</math> squares and <math>1</math> circle are intersected with each, there are <math>143 \cdot (3+1) = 572</math> squares and circles intersected. Adding the circle and square that are intersected at the origin back into the picture, we get that there are <math>572+2=\boxed{574}</math> squares and circles intersected in total.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2016|n=I|num-b=13|num-a=15}}
 
{{AIME box|year=2016|n=I|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 15:55, 5 March 2016

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 1

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 $(2 + \frac{1}{10}, 1 - \frac{1}{10}) = (\frac{21}{10}, \frac{9}{10})$. Because $\frac{9}{10} = \frac{3}{7} \cdot \frac{21}{10}$, $(\frac{21}{10}, \frac{9}{10})$ lies on $l$. Since the circle centered at $(2,1)$ is contained inside the square, this circle does not intersect $l$. Similarly the upper left vertex of the square centered at $(5,2)$ is on $l$. Since every other point listed above is farther away from a lattice point (excluding (0,0) and (7,3)) and there are two squares with centers strictly between $(0,0)$ and $(7,3)$ that intersect $l$. Since there are $\frac{1001}{7} = \frac{429}{3} = 143$ segments from $(7k, 3k)$ to $(7(k + 1), 3(k + 1))$, the above count is yields $143 \cdot 2 = 286$ squares. Since every lattice point on $l$ is of the form $(3k, 7k)$ where $0 \le k \le 143$, there are $144$ lattice points on $l$. Centered at each lattice point, there is one square and one circle, hence this counts $288$ squares and circles. Thus $m + n = 286 + 288 = \boxed{574}$.

(Solution by gundraja) [asy]size(12cm);draw((0,0)--(7,3));draw(box((0,0),(7,3)),dotted); for(int i=0;i<8;++i)for(int j=0;j<4;++j){dot((i,j),linewidth(1));draw(box((i-.1,j-.1),(i+.1,j+.1)),linewidth(.5));draw(circle((i,j),.1),linewidth(.5));} [/asy]

Solution 2

This is mostly a clarification since Solution 1 reads a little confusingly near the end, but let's take the diagram for the origin to $(7,3)$. We have the origin circle and square intersected, then two squares, then the circle and square at $(7,3)$. If we take the circle and square at the origin out of the diagram, we will be able to repeat the resulting segment (with its circles and squares) end to end from $(0,0)$ to $(1001,429)$, which forms the line we need without overlapping. Since $143$ of these segments are needed to do this, and $3$ squares and $1$ circle are intersected with each, there are $143 \cdot (3+1) = 572$ squares and circles intersected. Adding the circle and square that are intersected at the origin back into the picture, we get that there are $572+2=\boxed{574}$ squares and circles intersected in total.

See also

2016 AIME I (Problems • Answer Key • Resources)
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Problem 13 		Followed by
Problem 15
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