Difference between revisions of "2010 AMC 12A Problems/Problem 25"

Latest revision as of 14:00, 12 October 2024

Problem

Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?

$\textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$

Solution 1

It should first be noted that given any quadrilateral of fixed side lengths, there is exactly one way to manipulate the angles so that the quadrilateral becomes cyclic.

Proof. Given a quadrilateral $ABCD$ where all sides are fixed (in a certain order), we can construct the diagonal $\overline{BD}$ . When $BD$ is the minimum allowed by the triangle inequality, one of the angles $\angle DAB$ or $\angle BCD$ will be degenerate and measure $0^\circ$ , so opposite angles will sum to less than $180^\circ$ . When $BD$ is the maximum allowed, one of the angles will be degenerate and measure $180^\circ$ , so opposite angles will sum to more than $180^\circ$ . Thus, since the sum of opposite angles increases continuously as $BD$ is lengthened from the minimum to the maximum values, there is a unique value of $BD$ somewhere in the middle such that the sum of opposite angles is exactly $180^\circ$ .

Denote $a$ , $b$ , $c$ , and $d$ as the integer side lengths of the quadrilateral. Without loss of generality, let $a\ge b \ge c \ge d$ .

Since $a+b+c+d = 32$ , the Triangle Inequality implies that $a \le 15$ .

We will now split into $5$ cases.

Case $1$ : $a = b = c = d$ ( $4$ side lengths are equal)

Clearly there is only $1$ way to select the side lengths $(8,8,8,8)$ , and no matter how the sides are rearranged only $1$ unique quadrilateral can be formed.

Case $2$ : $a = b = c > d$ or $a > b = c = d$ ( $3$ side lengths are equal)

If $3$ side lengths are equal, then each of those side lengths can only be integers from $6$ to $10$ except for $8$ (because that is counted in the first case). Obviously there is still only $1$ unique quadrilateral that can be formed from one set of side lengths, resulting in a total of $4$ quadrilaterals.

Case $3$ : $a = b > c = d$ ( $2$ pairs of side lengths are equal)

$a$ and $b$ can be any integer from $9$ to $15$ , and likewise $c$ and $d$ can be any integer from $1$ to $7$ . However, a single set of side lengths can form $2$ different cyclic quadrilaterals (a rectangle and a kite), so the total number of quadrilaterals for this case is $7\cdot{2} = 14$ .

Case $4$ : $a = b > c > d$ or $a > b = c > d$ or $a > b > c = d$ ( $2$ side lengths are equal)

If the $2$ equal side lengths are each $1$ , then the other $2$ sides must each be $15$ , which we have already counted in an earlier case. If the equal side lengths are each $2$ , there is $1$ possible set of side lengths. Likewise, for side lengths of $3$ there are $2$ sets. Continuing this pattern, we find a total of $1+2+3+4+4+5+7+5+4+4+3+2+1 = 45$ sets of side lengths. (Be VERY careful when adding up the total for this case!) For each set of side lengths, there are $3$ possible quadrilaterals that can be formed, so the total number of quadrilaterals for this case is $3\cdot{45} = 135$ .

Case $5$ : $a > b > c > d$ (no side lengths are equal) Using the same counting principles starting from $a = 15$ and eventually reaching $a = 9$ , we find that the total number of possible side lengths is $69$ . There are $4!$ ways to arrange the $4$ side lengths, but there is only $1$ unique quadrilateral for $4$ rotations, so the number of quadrilaterals for each set of side lengths is $\frac{4!}{4} = 6$ . The total number of quadrilaterals is $6\cdot{69} = 414$ .

And so, the total number of quadrilaterals that can be made is $414 + 135 + 14 + 4 + 1 = \boxed{568\ \textbf{(C)}}$ .

Solution 2

As with solution $1$ we would like to note that given any quadrilateral we can change its angles to make a cyclic one.

Let $a \ge b \ge c\ge d$ be the sides of the quadrilateral.

There are $\binom{31}{3}$ ways to partition $32$ . However, some of these will not be quadrilaterals since they would have one side bigger than the sum of the other three. This occurs when $a \ge 16$ . For $a=16$ , $b+c+d=16$ . There are $\binom{15}{2}$ ways to partition $16$ . Since $a$ could be any of the four sides, we have counted $4\binom{15}{2}$ degenerate quadrilaterals. Similarly, there are $4\binom{14}{2}$ , $4\binom{13}{2} \cdots 4\binom{2}{2}$ for other values of $a$ . Thus, there are $\binom{31}{3} - 4\left(\binom{15}{2}+\binom{14}{2}+\cdots+\binom{2}{2}\right) = \binom{31}{3} - 4\binom{16}{3} = 2255$ non-degenerate partitions of $32$ by the hockey stick theorem. We then account for symmetry. If all sides are congruent (meaning the quadrilateral is a square), the quadrilateral will be counted once. If the quadrilateral is a rectangle (and not a square), it will be counted twice. In all other cases, it will be counted 4 times. Since there is $1$ square case, and $7$ rectangle cases, there are $2255-1-2\cdot7=2240$ quadrilaterals counted 4 times. Thus there are $1+7+\frac{2240}{4} = \boxed{568} = \boxed{\textbf{(C)}}$ total quadrilaterals.

Solution 3 (Burnside)

As with solution $1$ we find that there are $2255$ ways to form a quadrilateral if we don't account for rotations. We now apply Burnside's Lemma. There are four types of actions in the group acting on the set of quadrilaterals. We will consider each individually: Identity: maps a quadrilateral with sides $a,b,c,d$ in that order to $a,b,c,d$ . Obviously all members of the set of quadrilaterals are fixed points, for a total of $2255$ . Rotation by one: maps a quadrilateral from $a,b,c,d$ to $b,c,d,a$ . For this to have a fixed point we need $a=b,b=c,c=d,d=a$ , so the only quadrilateral that is a fixed point is the square with side length $8$ , for a total of $1$ . Rotation by two: maps a quadrilateral from $a,b,c,d$ to $c,d,a,b$ . For this to be a fixed point we need $a=c$ and $b=d$ . Thus the quadrilateral is of the form $x,y,x,y$ —a rectangle. We can count that there are $15$ rectangle cases, namely $(a, b, c, d) = \{ (1, 15, 1, 15), (2, 14, 2, 14), \cdots, (15, 1, 15, 1) \}$ . Rotation by three: maps a quadrilateral from $a,b,c,d$ to $d,a,b,c$ . Similarly to the rotation by one case, there is one fixed point here. Summing up, we get that the total number of groups is $2255+1+15+1=2272$ . Since there are $4$ members of the group our final answer is $\frac{2272}{4}=\boxed{568}$ total orbits of the set of quadrilaterals, so the answer is $\boxed{\textbf{C}}$ .

@@ Line 42: / Line 42: @@
 As with solution <math>1</math> we would  like to note that given any quadrilateral we can change its angles to make a cyclic one.
-Let <math>a\ge b\ge c\ge d</math> be the sides of the quadrilateral.
+Let <math>a \ge b \ge c\ge d</math> be the sides of the quadrilateral.
-There are <math>\binom{31}{3}</math> ways to partition <math>32</math>. However, some of these will not be quadrilaterals since they would have one side bigger than the sum of the other three.  This occurs when <math>a \ge 16</math>.  For <math>a=16</math>, <math>b+c+d=16</math>. There are <math>\binom{15}{2}</math> ways to partition <math>16</math>. Since <math>a</math> could be any of the four sides, we have counted <math>4\binom{15}{2}</math> degenerate quadrilaterals. Similarly, there are <math>4\binom{14}{2}</math>, <math>4\binom{13}{2} \cdots 4\binom{2}{2}</math> for other values of <math>a</math>.  Thus, there are <math>\binom{31}{3} - 4\left(\binom{15}{2}+\binom{14}{2}+\cdots+\binom{2}{2}\right) = \binom{31}{3} - 4\binom{16}{3} = 2255</math> non-degenerate partitions of <math>32</math> by the hockey stick theorem.  However, for <math>a\neq b \neq c \neq d</math> or <math>a=b \neq c \neq d</math>, each quadrilateral is counted <math>4</math> times, <math>1</math> for each rotation. Also, for <math>a=b\neq c=d</math>, each quadrilateral is counted twice. Since there is <math>1</math> quadrilateral for which <math>a=b=c=d</math>, and <math>7</math> for which <math>a=b\neq  c=d</math>, there are <math>2255-1-2\cdot7=2240</math> quads for which <math>a\neq b \neq c \neq d</math> or <math>a=b \neq c \neq d</math>. Thus there are <math>1+7+\frac{2240}{4} = \boxed{568} = \boxed{\textbf{(C)}}</math> total quadrilaterals.
+There are <math>\binom{31}{3}</math> ways to partition <math>32</math>. However, some of these will not be quadrilaterals since they would have one side bigger than the sum of the other three.  This occurs when <math>a \ge 16</math>.  For <math>a=16</math>, <math>b+c+d=16</math>. There are <math>\binom{15}{2}</math> ways to partition <math>16</math>. Since <math>a</math> could be any of the four sides, we have counted <math>4\binom{15}{2}</math> degenerate quadrilaterals. Similarly, there are <math>4\binom{14}{2}</math>, <math>4\binom{13}{2} \cdots 4\binom{2}{2}</math> for other values of <math>a</math>.  Thus, there are <math>\binom{31}{3} - 4\left(\binom{15}{2}+\binom{14}{2}+\cdots+\binom{2}{2}\right) = \binom{31}{3} - 4\binom{16}{3} = 2255</math> non-degenerate partitions of <math>32</math> by the hockey stick theorem.  We then account for symmetry. If all sides are congruent (meaning the quadrilateral is a square), the quadrilateral will be counted once. If the quadrilateral is a rectangle (and not a square), it will be counted twice. In all other cases, it will be counted 4 times. Since there is <math>1</math> square case, and <math>7</math> rectangle cases, there are <math>2255-1-2\cdot7=2240</math> quadrilaterals counted 4 times. Thus there are <math>1+7+\frac{2240}{4} = \boxed{568} = \boxed{\textbf{(C)}}</math> total quadrilaterals.
+==Solution 3 (Burnside)==
+As with solution <math>1</math> we find that there are <math>2255</math> ways to form a quadrilateral if we don't account for rotations. We now apply [[Burnside's Lemma]]. There are four types of actions in the group acting on the set of quadrilaterals. We will consider each individually:
+Identity: maps a quadrilateral with sides <math>a,b,c,d</math> in that order to <math>a,b,c,d</math>. Obviously all members of the set of quadrilaterals are fixed points, for a total of <math>2255</math>.
+Rotation by one: maps a quadrilateral from <math>a,b,c,d</math> to <math>b,c,d,a</math>. For this to have a fixed point we need <math>a=b,b=c,c=d,d=a</math>, so the only quadrilateral that is a fixed point is the square with side length <math>8</math>, for a total of <math>1</math>.
+Rotation by two: maps a quadrilateral from <math>a,b,c,d</math> to <math>c,d,a,b</math>. For this to be a fixed point we need <math>a=c</math> and <math>b=d</math>. Thus the quadrilateral is of the form <math>x,y,x,y</math>—a rectangle. We can count that there are <math>15</math> rectangle cases, namely <math>(a, b, c, d) = \{ (1, 15, 1, 15), (2, 14, 2, 14), \cdots, (15, 1, 15, 1)  \}</math>.
+Rotation by three: maps a quadrilateral from <math>a,b,c,d</math> to <math>d,a,b,c</math>. Similarly to the rotation by one case, there is one fixed point here.
+Summing up, we get that the total number of groups is <math>2255+1+15+1=2272</math>. Since there are <math>4</math> members of the group our final answer is <math>\frac{2272}{4}=\boxed{568}</math> total orbits of the set of quadrilaterals, so the answer is <math>\boxed{\textbf{C}}</math>.
 == See also ==

2010 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last Problem
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
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