Difference between revisions of "2015 AMC 12B Problems/Problem 23"

Latest revision as of 02:15, 21 October 2024

The following problem is from both the 2015 AMC 10B #25 and 2015 AMC 12B #23, so both problems redirect to this page.

Problem

A rectangular box measures $a \times b \times c$ , where $a$ , $b$ , and $c$ are integers and $1\leq a \leq b \leq c$ . The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?

$\textbf{(A)}\; 4 \qquad\textbf{(B)}\; 10 \qquad\textbf{(C)}\; 12 \qquad\textbf{(D)}\; 21 \qquad\textbf{(E)}\; 26$

Solution 1

We need $abc = 2(ab+bc+ac) \quad \text{ or } \quad (a-2)bc = 2a(b+c).$ Since $a\le b$ and $a,b,c$ are all positive $,ac \le bc$ . From the first equation we get $abc \le 6bc$ . Thus $a\le 6$ . From the second equation we see that $a > 2$ . Thus $a\in \{3, 4, 5, 6\}$ . For the following we need the resulting $(b,c)$ to be positive integers and $b$ and $c$ to satisfy the condition $1\leq a \leq b \leq c.$

If $a=3$ we need $bc = 6(b+c) \Rightarrow (b-6)(c-6)=36$ . We get five roots $\{(3, 7, 42), (3, 8, 24), (3,9,18), (3, 10, 15), (3,12,12)\}$ .

If $a=4$ we need $2bc = 8(b+c) \Rightarrow bc = 4(b+c) \Rightarrow (b-4)(c-4)=16$ . We get three roots $\{(4,5,20), (4,6,12), (4,8,8)\}$ .

If $a=5$ we need $3bc = 10(b+c) \Rightarrow 9bc=30(b+c) \Rightarrow (3b-10)(3c-10)=100$ . We get one root $\{(5,5,10)\}$ .

If $a=6$ we need $4bc = 12(b+c) \Rightarrow bc = 3(b+c) \Rightarrow (b-3)(c-3)=9$ . We get one root $\{(6,6,6)\}$ .

Thus, there are $5+3+1+1 = \boxed{\textbf{(B)}\; 10}$ solutions.

Solution 2

The surface area is $2(ab+bc+ca)$ , and the volume is $abc$ , so equating the two yields

$2(ab+bc+ca)=abc.$

Divide both sides by $2abc$ to obtain $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}.$

First consider the bound of the variable $a$ . Since $\frac{1}{a}<\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2},$ we have $a>2$ , or $a\geqslant3$ .

Also note that $c \geq b \geq a > 0$ , hence $\frac{1}{a} \geq \frac{1}{b} \geq \frac{1}{c}$ . Thus, $\frac{1}{2}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \leq \frac{3}{a}$ , so $a \leq 6$ .

So we have $a=3, 4, 5$ or $6$ .

Before the casework, let's consider the possible range for $b$ if $\frac{1}{b}+\frac{1}{c}=k>0$ . From $\frac{1}{b}<k$ , we have $b>\frac{1}{k}$ . From $\frac{2}{b} \geq \frac{1}{b}+\frac{1}{c}=k$ , we have $b \leq \frac{2}{k}$ . Thus $\frac{1}{k}<b \leq \frac{2}{k}$ .

When $a=3$ , we get $\frac{1}{b}+\frac{1}{c}=\frac{1}{6}$ , so $b=7, 8, 9, 10, 11, 12$ . We find the solutions $(a, b, c)=(3, 7, 42)$ , $(3, 8, 24)$ , $(3, 9, 18)$ , $(3, 10, 15)$ , $(3, 12, 12)$ , for a total of $5$ solutions.

When $a=4$ , we get $\frac{1}{b}+\frac{1}{c}=\frac{1}{4}$ , so $b=5, 6, 7, 8$ . We find the solutions $(a, b, c)=(4, 5, 20)$ , $(4, 6, 12)$ , $(4, 8, 8)$ , for a total of $3$ solutions.

When $a=5$ , we get $\frac{1}{b}+\frac{1}{c}=\frac{3}{10}$ , so $b=5, 6$ . The only solution in this case is $(a, b, c)=(5, 5, 10)$ .

When $a=6$ , $b$ is forced to be $6$ , and thus $(a, b, c)=(6, 6, 6)$ .

Thus, there are $5+3+1+1 = \boxed{\textbf{(B)}\; 10}$ solutions.

Simplification of Solution 2

The surface area is $2(ab+bc+ca)$ , the volume is $abc$ , so $2(ab+bc+ca)=abc$ .

Divide both sides by $2abc$ , we have: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}.$ First consider the bound of the variable $a$ . Since $\frac{1}{a}<\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2},$ we have $a>2$ , or $a\ge 3$ .

Also note that $c\ge b\ge a>0$ , we have $\frac{1}{a}\ge \frac{1}{b}\ge \frac{1}{c}$ . Thus, $\frac{1}{2}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le \frac{3}{a}$ , so $a\le 6$ .

So we have $a=3, 4, 5$ or $6$ .

We can say $\frac{1}{b}+\frac{1}{c}=\frac{1}{q}$ , where $\frac{1}{q} = \frac{1}{2}-\frac{1}{a}$ .

Notice $\emph{\text{immediately}}$ that $b, c > q$ . This is our key step. Then we can say $b=q+d$ , $c=q+e$ . If we clear the fraction about b and c (do the math), our immediate result is that $de = q^2$ . Realize also that $d \leq e$ .

Now go through cases for $a$ and you end up with the same result. However, now you don't have to guess solutions. For example, when $a=3$ , then $de = 36$ and $d=1, 2, 3, 4, 6$ .

Video Solution

https://www.youtube.com/watch?v=JFUpe32aWnw&t=1941s

@@ Line 10: / Line 10: @@
 We need <cmath>abc = 2(ab+bc+ac) \quad \text{ or } \quad (a-2)bc = 2a(b+c).</cmath>
-Since <math>a\le b</math> and <math>a,b,c</math> are all positive<math>,ac \le bc</math>. From the first equation we get <math>abc \le 6bc</math>. Thus <math>a\le 6</math>. From the second equation we see that <math>a > 2</math>. Thus <math>a\in \{3, 4, 5, 6\}</math>.
+Since <math>a\le b</math> and <math>a,b,c</math> are all positive<math>,ac \le bc</math>. From the first equation we get <math>abc \le 6bc</math>. Thus <math>a\le 6</math>. From the second equation we see that <math>a > 2</math>. Thus <math>a\in \{3, 4, 5, 6\}</math>. For the following we need the resulting <math>(b,c)</math> to be positive integers and <math>b</math> and <math>c</math> to satisfy the condition <math>1\leq a \leq b \leq c.</math>
 *If <math>a=3</math> we need <math>bc = 6(b+c) \Rightarrow (b-6)(c-6)=36</math>. We get '''five''' roots <math>\{(3, 7, 42), (3, 8, 24), (3,9,18), (3, 10, 15), (3,12,12)\}</math>.

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