Difference between revisions of "2015 AMC 10B Problems/Problem 25"
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- minor edit by Williamgolly, minor edit by Tiblis | - minor edit by Williamgolly, minor edit by Tiblis | ||
+ | |||
+ | Solution | ||
+ | We need<cmath>abc = 2(ab+bc+ac) \quad \text{ or } \quad (a-2)bc = 2a(b+c).</cmath>Since <math>ab, ac \le bc</math>, 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>. | ||
+ | |||
+ | 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> | ||
+ | If <math>a=4</math> we need <math>bc = 4(b+c) \Rightarrow (b-4)(c-4)=16</math>. We get three roots <math>\{(4,5,20), (4,6,12), (4,8,8)\}</math>. | ||
+ | If <math>a=5</math> we need <math>3bc = 10(b+c)</math>, which is the same as <math>9bc=30(b+c)\Rightarrow (3b-10)(3c-10)=100</math>. We get only one root (corresponding to <math>100=5\cdot 20</math>) <math>(5,5,10)</math>. | ||
+ | If <math>a=6</math> we need <math>4bc = 12(b+c)</math>. Then <math>(b-3)(c-3)=9</math>. We get one root <math>(6,6,6)</math>. | ||
+ | Thus, there are <math>5+3+1+1 = \boxed{\textbf{(B)}\; 10}</math> solutions. | ||
+ | |||
+ | Solution 2 | ||
+ | The surface area is <math>2(ab+bc+ca)</math>, and the volume is <math>abc</math>, so equating the two yields | ||
+ | |||
+ | <cmath>2(ab+bc+ca)=abc.</cmath> | ||
+ | Divide both sides by <math>2abc</math> to obtain<cmath>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}.</cmath> | ||
+ | First consider the bound of the variable <math>a</math>. Since <math>\frac{1}{a}<\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2},</math> we have <math>a>2</math>, or <math>a\geqslant3</math>. | ||
+ | |||
+ | Also note that <math>c \geq b \geq a > 0</math>, hence <math>\frac{1}{a} \geq \frac{1}{b} \geq \frac{1}{c}</math>. Thus, <math>\frac{1}{2}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \leq \frac{3}{a}</math>, so <math>a \leq 6</math>. | ||
+ | |||
+ | So we have <math>a=3, 4, 5</math> or <math>6</math>. | ||
+ | |||
+ | Before the casework, let's consider the possible range for <math>b</math> if <math>\frac{1}{b}+\frac{1}{c}=k>0</math>. From <math>\frac{1}{b}<k</math>, we have <math>b>\frac{1}{k}</math>. From <math>\frac{2}{b} \geq \frac{1}{b}+\frac{1}{c}=k</math>, we have <math>b \leq \frac{2}{k}</math>. Thus <math>\frac{1}{k}<b \leq \frac{2}{k}</math>. | ||
+ | |||
+ | When <math>a=3</math>, we get <math>\frac{1}{b}+\frac{1}{c}=\frac{1}{6}</math>, so <math>b=7, 8, 9, 10, 11, 12</math>. We find the solutions <math>(a, b, c)=(3, 7, 42)</math>, <math>(3, 8, 24)</math>, <math>(3, 9, 18)</math>, <math>(3, 10, 15)</math>, <math>(3, 12, 12)</math>, for a total of <math>5</math> solutions. | ||
+ | |||
+ | When <math>a=4</math>, we get <math>\frac{1}{b}+\frac{1}{c}=\frac{1}{4}</math>, so <math>b=5, 6, 7, 8</math>. We find the solutions <math>(a, b, c)=(4, 5, 20)</math>, <math>(4, 6, 12)</math>, <math>(4, 8, 8)</math>, for a total of <math>3</math> solutions. | ||
+ | |||
+ | When <math>a=5</math>, we get <math>\frac{1}{b}+\frac{1}{c}=\frac{3}{10}</math>, so <math>b=5, 6</math>. The only solution in this case is <math>(a, b, c)=(5, 5, 10)</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>5+3+1+1 = \boxed{\textbf{(B)}\; 10}</math> solutions. | ||
==See Also== | ==See Also== |
Revision as of 20:59, 5 July 2020
Problem
A rectangular box measures , where
,
, and
are integers and
. The volume and the surface area of the box are numerically equal. How many ordered triples
are possible?
Solution
The surface area is , the volume is
, so
.
Divide both sides by , we have:
First consider the bound of the variable . Since
we have
, or
.
Also note that , we have
.
Thus,
, so
.
So we have or
.
Before the casework, let's consider the possible range for if
.
From , we have
. From
, we have
. Thus
When ,
, so
. The solutions we find are
, for a total of
solutions.
When ,
, so
. The solutions we find are
, for a total of
solutions.
When ,
, so
. The only solution in this case is
.
When ,
is forced to be
, and thus
.
Thus, our answer is
Simplification of Solution
The surface area is , the volume is
, so
.
Divide both sides by , we have:
First consider the bound of the variable
. Since
we have
, or
.
Also note that , we have
. Thus,
, so
.
So we have or
.
We can say , where
.
Notice that
This is our key step.
Then we can say
,
. If we clear the fraction about b and c (do the math), our immediate result is that
. Realize also that
.
Now go through cases for and you end up with the same result. However, now you don't have to guess solutions. For example, when
, then
and
.
- minor edit by Williamgolly, minor edit by Tiblis
Solution
We needSince
, we get
. Thus
. From the second equation we see that
. Thus
.
If we need
. We get five roots
If
we need
. We get three roots
.
If
we need
, which is the same as
. We get only one root (corresponding to
)
.
If
we need
. Then
. We get one root
.
Thus, there are
solutions.
Solution 2
The surface area is , and the volume is
, so equating the two yields
Divide both sides by
to obtain
First consider the bound of the variable
. Since
we have
, or
.
Also note that , hence
. Thus,
, so
.
So we have or
.
Before the casework, let's consider the possible range for if
. From
, we have
. From
, we have
. Thus
.
When , we get
, so
. We find the solutions
,
,
,
,
, for a total of
solutions.
When , we get
, so
. We find the solutions
,
,
, for a total of
solutions.
When , we get
, so
. The only solution in this case is
.
When ,
is forced to be
, and thus
.
Thus, there are solutions.
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Question | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.