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Difference between revisions of "2011 AMC 12A Problems/Problem 22"

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== Solution ==
 
== Solution ==
  
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First, notice that there must be four rays emanating from <math>X</math> that intersect the four corners of the square region. Depending on the location of <math>X</math>, the number of rays distributed among these four triangular sectors will vary. We start by finding the corner-most point that is <math>100</math>-ray partitional. We first draw the four rays that intersect the vertices. At this point, the triangular sectors with bases as the sides of the square that the point is closest to both do not have rays dividing their areas. Therefore, their heights are equivalent since their areas are equal. The remaining <math>96</math> rays are divided among the other two triangular sectors, each sector with <math>48</math> rays, thus dividing these two sectors into <math>49</math> triangles of equal areas. Let the distance from this corner point to the closest side be <math>a</math> and the side of the square be <math>s</math>. From this, we get the equation <math>\frac{a\times s}{2}=\frac{(1-a)\times s}{2}\times\frac1{49}</math>.
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First, notice that there must be four rays emanating from <math>X</math> that intersect the four corners of the square region. Depending on the location of <math>X</math>, the number of rays distributed among these four triangular sectors will vary. We start by finding the corner-most point that is <math>100</math>-ray partitional (let this point be the bottom-left-most point). We first draw the four rays that intersect the vertices. At this point, the triangular sectors with bases as the sides of the square that the point is closest to both do not have rays dividing their areas. Therefore, their heights are equivalent since their areas are equal. The remaining <math>96</math> rays are divided among the other two triangular sectors, each sector with <math>48</math> rays, thus dividing these two sectors into <math>49</math> triangles of equal areas. Let the distance from this corner point to the closest side be <math>a</math> and the side of the square be <math>s</math>. From this, we get the equation <math>\frac{a\times s}{2}=\frac{(1-a)\times s}{2}\times\frac1{49}</math>. Solve for <math>a</math> to get <math>a=\frac s{50}</math>. Therefore, point <math>X</math> is <math>\frac1{50}</math> of the side length away from the two sides it is closest to. By moving <math>X</math> <math>\frac s{50}</math> to the right, we also move one ray from the right sector to the left sector, which determines another <math>100</math>-ray partitional point. We can continue moving <math>X</math> right and up to derive the set of points that are <math>100</math>-ray partitional. In the end, we get a square grid of points each <math>\frac s{50}</math> apart from one another. Since this grid ranges from a distance of <math>\frac s{50}</math> from one side to <math>\frac{49s}{50}</math> from the same side, we have a <math>49\times49</math> grid, a total of <math>2401</math> <math>100</math>-ray partitional points. To find the overlap from the <math>60</math>-ray partitional, we must find the distance from the corner-most <math>60</math>-ray partitional point to the sides closest to it. Since the <math>100</math>-ray partitional points form a <math>49\times49</math> grid, each point <math>\frac s{50}</math> apart from each other, we can deduce that the <math>60</math>-ray partitional points form a <math>29\times29</math> grid, each point <math>\frac s{30}</math> apart from each other. To find the overlap points, we must find the common divisors of <math>30</math> and <math>50</math> which are <math>1, 2, 5,</math> and <math>10</math>. Therefore, the overlapping points will form grids with points <math>s</math>, <math>\frac s{2}</math>, <math>\frac s{5}</math>, and <math>\frac s{10}</math> away from each other respectively. Since the grid with points <math>\frac s{10}</math> away from each other includes the other points, we can disregard the other grids. The total overlapping set of points is a <math>9\times9</math> grid, which has <math>81</math> points. Subtract <math>81</math> from <math>2401</math> to get <math>2401-81=\boxed{\textbf{(C)}\ 2320}</math>.
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2011|num-b=21|num-a=23|ab=A}}
 
{{AMC12 box|year=2011|num-b=21|num-a=23|ab=A}}

Revision as of 23:20, 12 February 2011

Problem

Let $R$ be a square region and $n \geq 4$ an integer. A point $X$ in the interior or $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$-ray partitional but not $60$-ray partitional?

$\textbf{(A)}\ 1500 \qquad \textbf{(B)}\ 1560 \qquad \textbf{(C)}\ 2320 \qquad \textbf{(D)}\ 2480 \qquad \textbf{(E)}\ 2500$

Solution

First, notice that there must be four rays emanating from $X$ that intersect the four corners of the square region. Depending on the location of $X$, the number of rays distributed among these four triangular sectors will vary. We start by finding the corner-most point that is $100$-ray partitional (let this point be the bottom-left-most point). We first draw the four rays that intersect the vertices. At this point, the triangular sectors with bases as the sides of the square that the point is closest to both do not have rays dividing their areas. Therefore, their heights are equivalent since their areas are equal. The remaining $96$ rays are divided among the other two triangular sectors, each sector with $48$ rays, thus dividing these two sectors into $49$ triangles of equal areas. Let the distance from this corner point to the closest side be $a$ and the side of the square be $s$. From this, we get the equation $\frac{a\times s}{2}=\frac{(1-a)\times s}{2}\times\frac1{49}$. Solve for $a$ to get $a=\frac s{50}$. Therefore, point $X$ is $\frac1{50}$ of the side length away from the two sides it is closest to. By moving $X$ $\frac s{50}$ to the right, we also move one ray from the right sector to the left sector, which determines another $100$-ray partitional point. We can continue moving $X$ right and up to derive the set of points that are $100$-ray partitional. In the end, we get a square grid of points each $\frac s{50}$ apart from one another. Since this grid ranges from a distance of $\frac s{50}$ from one side to $\frac{49s}{50}$ from the same side, we have a $49\times49$ grid, a total of $2401$ $100$-ray partitional points. To find the overlap from the $60$-ray partitional, we must find the distance from the corner-most $60$-ray partitional point to the sides closest to it. Since the $100$-ray partitional points form a $49\times49$ grid, each point $\frac s{50}$ apart from each other, we can deduce that the $60$-ray partitional points form a $29\times29$ grid, each point $\frac s{30}$ apart from each other. To find the overlap points, we must find the common divisors of $30$ and $50$ which are $1, 2, 5,$ and $10$. Therefore, the overlapping points will form grids with points $s$, $\frac s{2}$, $\frac s{5}$, and $\frac s{10}$ away from each other respectively. Since the grid with points $\frac s{10}$ away from each other includes the other points, we can disregard the other grids. The total overlapping set of points is a $9\times9$ grid, which has $81$ points. Subtract $81$ from $2401$ to get $2401-81=\boxed{\textbf{(C)}\ 2320}$.

See also

2011 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions
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