AMC 10 · 2025 · #7

Grade 11 algebra
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Multiplicative Rate or Series
logarithm-propertiesexponentsformula-substitution convert-to-algebra ↑ Prerequisites: logarithm-propertiesexponents
📏 Medium solution 💡 2 insights
Problem
An organism's top speed follows the model v = kn^am^b, where n is its number of toes, m is its number of eyes, and k, a, b are integer constants. For a group with a fixed 5 toes, log v = 4 + 2 log m; for a group with a fixed 25 eyes, log v = 4 + 4 log n (base 10). Find k + a + b.

Pick an answer.

(A)
20
(B)
21
(C)
22
(D)
23
(E)
24
How to solve
Strategy Convert to Algebra

The model multiplies powers together, and the two clues are written with logarithms. Taking the log of the model turns the product into a sum, so it lines up term-by-term with the given log equations. Once both sides are in the same shape, the exponents and the constant just have to match, which reads off a, b, and then k without any guessing.

1STEP 1

Take the log of the model

Take base-10 log: log v = log k + a·log n + b·log m. The product of powers becomes a sum, so a, b are the coefficients of log n, log m.

log v = log(k n^am^b) = log k + alog n + blog m
2STEP 2

Match the 5-toe population

With n = 5 fixed, log k + a·log 5 is a constant, so (const) + b·log m = 4 + 2·log m forces b = 2 and log k + a·log 5 = 4.

(log k + alog 5) + blog m = 4 + 2log m → b = 2, log k + alog 5 = 4
3STEP 3

Match the 25-eye population

With m = 25 fixed, log k + b·log 25 is a constant, so (const) + a·log n = 4 + 4·log n forces a = 4 and log k + b·log 25 = 4.

(log k + blog 25) + alog n = 4 + 4log n → a = 4, log k + blog 25 = 4
4STEP 4

Solve for k

Put a = 4 into log k + a·log 5 = 4, i.e. k·5⁴ = 10⁴. With 5⁴ = 625, k = 10000625\frac{10000}{625} = 16, an integer as required.

log k + 4log 5 = 4 → k · 5⁴ = 10⁴ → k = 10000/625 = 16
5STEP 5

Check consistency and add

Check k = 16 also fits the eyes: 16·25² = 10000 = 10⁴. So k, a, b = 16, 4, 2 and k + a + b = 22.

k + a + b = 16 + 4 + 2 = 22
Answer
22
Rebuild the model with k = 16, a = 4, b = 2: v = 16n⁴m². For n = 5, v = 16625m² = 10000m², so log v = log(10⁴ m²) = 4 + 2 log m, matching the first clue. For m = 25, v = 16n⁴625 = 10000n⁴, so log v = 4 + 4 log n, matching the second clue. Both given equations hold, and every constant is an integer, so the solution is consistent.
💡Key takeaway

Taking a logarithm turns a product of powers into a sum, so you can line it up against the given equation and match the pieces one at a time.

  • Take the log of the model
  • Match the 5-toe population
  • Match the 25-eye population
  • Solve for k
  • Check consistency and add

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