AMC 10 · 2004 · #6

Grade 8 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

exponentsfactors identify-subproblemseasier-related-problem ↑ Prerequisites: exponents

📏 Medium solution 💡 2 insights

Problem

Six enormous numbers are built from powers of 2004. Five differences are formed from neighbouring pairs in that list. Decide which of the five differences is the largest.

Pick an answer.

(A)

U-V

(B)

V-W

(C)

W-X

(D)

X-Y

(E)

Y-Z

How to solve

Strategy Organize Information in More Ways

The six numbers look unrelated because they carry three different exponents (2005, 2004, 2003) and three different front coefficients (2, 2003, 1). Tool #15 fixes that: rewrite all six on one common scale, as a coefficient times 2004²⁰⁰³. Then every difference is just a difference of coefficients, and comparing five ordinary whole numbers replaces comparing five astronomically large ones. Tool #7 handles the five differences one at a time, and Tool #9 gives a small model with 2004 replaced by 5 to confirm the ranking is a property of the structure and not of the arithmetic.

1STEP 1

Put all six numbers on one scale

Every number carries the same power, so pull it out and put all six on one scale.

U=2·2004² u, V=2004² u, W=2003·2004 u, X=2·2004 u, Y=2004 u, Z=u where u=2004²⁰⁰³

2STEP 2

Subtract inside each pair

Each difference keeps that unit, so only the coefficients subtract.

U-V=2004²u, V-W=2004u, W-X=2004 · 2001 u, X-Y=2004u, Y-Z=2003u

3STEP 3

Compare the five coefficients

Only two coefficients compete, and they share a factor, so the larger is U minus V.

2004²=4,016,016 ＞ 2004·2001=4,010,004 ＞ 2004=2004 ＞ 2003

4STEP 4

Confirm on a shrunken model

Rebuilding the same shape with a small base confirms the same ranking, choice (A).

n=5: U-V=125, V-W=25, W-X=50, X-Y=25, Y-Z=20 → U-V largest → (A)

Answer

U-V

A size argument gives the same verdict without any of the coefficient arithmetic. Since U=2V, the difference U-V equals V=2004²⁰⁰⁵ itself — the biggest number in the whole problem. Meanwhile W, X, Y, Z are all at most W=2003 · 2004²⁰⁰⁴, so the three differences W-X, X-Y, Y-Z are each smaller than 2003 · 2004²⁰⁰⁴ ＜ 2004 · 2004²⁰⁰⁴=2004²⁰⁰⁵, and the remaining one works out to V-W=2004²⁰⁰⁴, smaller still. No other difference can reach 2004²⁰⁰⁵, so U-V must be the largest.

💡Key takeaway

Rewrite every giant number as a coefficient times the same power of 2004, and the contest between monsters becomes a contest between small numbers.

Put all six numbers on one scale
Subtract inside each pair
Compare the five coefficients
Confirm on a shrunken model

Put all six numbers on one scale

Subtract inside each pair

Compare the five coefficients

Confirm on a shrunken model

More like this