AMC 10 · 2015 · #25

Grade 7 geometry-3d

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

volume-rectangular-prismsurface-arealinear-diophantine bound-inequality-then-enumerate ↑ Prerequisites: surface-areavolume-rectangular-prism

📏 Long solution 💡 4 insights

Problem

A box has whole-number edges a ≤ b ≤ c. Its volume abc equals its surface area 2(ab+bc+ca) as a number. Count how many ordered triples (a,b,c) make this happen.

Pick an answer.

(A)

$\; 4$

(B)

$\; 10$

(C)

$\; 12$

(D)

$\; 21$

(E)

$\; 26$

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The question asks "how many", so the goal is a complete, no-overlap count (Tool #2, Make a Systematic List). To make the list finite, first translate volume and surface area into the equation abc=2(ab+bc+ca) and divide to get frac1a+frac1b+frac1c=1/2 (Tool #4, Introduce a Variable). The Extreme Principle (Tool #14) then traps the smallest edge a in a tiny range {3,4,5,6}, and each value of a becomes a separate subproblem (Tool #7) solved by listing factor pairs.

1STEP 1

Write the volume

A box with edges a, b, c has volume equal to the product abc.

V = abc

2STEP 2

Write the surface area and equate

Three pairs of identical faces give surface area 2(ab+bc+ca); set it equal to the volume abc.

abc = 2(ab+bc+ca)

3STEP 3

Divide by abc

Divide both sides by abc; each right-hand term drops one variable, leaving 1/a+1/b+1/c=1/2.

1 = 2(1/c+1/a+1/b) → 1/a+1/b+1/c=1/2

4STEP 4

Bound the smallest edge a

Since a carries the largest fraction, 1/2 ≤ 3/a and 1/a stays under 1/2, trapping a in {3, 4, 5, 6}.

1/2 ≤ 3/a → a ≤ 6, 1/a ＜ 1/2 → a ≥ 3

5STEP 5

Case a = 3

With a=3, 1/b+1/c=1/6 factors to (b-6)(c-6)=36; its factor pairs give 5 triples.

1/b+1/c=1/6 → (b-6)(c-6)=36

6STEP 6

Case a = 4

With a=4, 1/b+1/c=1/4 factors to (b-4)(c-4)=16; its factor pairs give 3 triples.

1/b+1/c=1/4 → (b-4)(c-4)=16

7STEP 7

Case a = 5

With a=5, 1/b+1/c=3/10 forces b=5 and c=10, giving the single triple (5,5,10).

1/b+1/c=3/10 → (b,c)=(5,10)

8STEP 8

Case a = 6

With a=6, 1/b+1/c=1/3 forces b=6 and c=6, giving the single cube (6,6,6).

1/b+1/c=1/3 → (b,c)=(6,6)

9STEP 9

Add the cases

The four cases never overlap, so add: 5+3+1+1=10 ordered triples, giving (B).

5 + 3 + 1 + 1 = 10 → (B)

Answer

10 is one of the listed choices. The cube (6,6,6) checks out directly: its volume is 216 and its surface area is 6 · 6² = 216, equal as required. The smallest edge a stayed inside {3,4,5,6}, so no case was skipped, and the counts 5,3,1,1 all came from complete factor-pair lists. The total 10 matches choice (B).

💡Key takeaway

Turn "volume equals surface area" into frac1a+frac1b+frac1c=1/2, squeeze the smallest edge into a few values, then count factor pairs in each case.

Write the volume
Write the surface area and equate
Divide by abc
Bound the smallest edge a
Case a = 3
Case a = 4
Case a = 5
Case a = 6
Add the cases