Question

Let $m$ be the smallest integer whose cube root has the form $n+r,$ where $n$ is a positive integer and $0<r<\frac{1}{1000}.$ Find $n.$

Answer

Answer: 19

Concepts:perfect power estimation inequality

Difficulty rating: 2110

Small Hint:

For fixed $n,$ the smallest possible integer is $m=n^3+1$

Big Hint:

Compare $n^3+1$ with $(n+0.001)^3$

Solution:

For a given $n,$ the closest integer cube-root candidate above $n$ is $m=n^3+1.$ We need $n^3+1<(n+0.001)^3,$ or $1<0.003n^2+0.000003n+10^{-9}.$ This fails at $n=18$ and holds at $n=19.$ The right-hand side is increasing for positive $n,$ so every smaller $n$ fails; every larger $n$ has a larger least candidate $m=n^3+1.$ Hence the smallest $m$ occurs with $n=19.$

1987 AIME Problem 12

Answer: 19

Difficulty rating: 2110

Small Hint:

Big Hint:

Solution:

Problem 12 in Other Years