Difference between revisions of "2015 AMC 10B Problems/Problem 25"

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When <math>a=6</math>, <math>b</math> is forced to be <math>6</math>, and thus <math>(a, b, c)=(6, 6, 6)</math>.
 
When <math>a=6</math>, <math>b</math> is forced to be <math>6</math>, and thus <math>(a, b, c)=(6, 6, 6)</math>.
  
Thus, there are <math>\boxed{\textbf{(B)}\;10}</math> solutions
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Thus, our answer is <math>\boxed{\textbf{(B)}\;10}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2015|ab=B|after=Last Problem|num-b=24}}
 
{{AMC10 box|year=2015|ab=B|after=Last Problem|num-b=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 02:00, 4 March 2015

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

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

Divide both sides by $2abc$, we get that \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}.\]

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 $c\geqslant b\geqslant a>0$, we have $\frac{1}{a}>\frac{1}{b}>\frac{1}{c}$. Thus, $\frac{1}{2}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geqslant \frac{3}{a}$, so $a\leqslant6$.

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 $\frac{2}{b}\geqslant\frac{1}{b}+\frac{1}{c}=k$, we have $b\leqslant\frac{2}{k}$. Thus $\frac{1}{k}<b\leqslant\frac{2}{k}$

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

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

When $a=5$, $\frac{1}{b}+\frac{1}{c}=\frac{3}{10}$, 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, our answer is $\boxed{\textbf{(B)}\;10}$

See Also

2015 AMC 10B (ProblemsAnswer KeyResources)
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
All AMC 10 Problems and Solutions

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