MathDB

Floor of cube roots

Source: AIME II 2007 #7

March 29, 2007

geometry3D geometryfloor functioncalculusintegrationAMC

Problem Statement

Given a real number

x,

let

\lfloor x \rfloor

denote the greatest integer less than or equal to

x.

For a certain integer

k,

there are exactly

70

positive integers

n_{1}, n_{2}, \ldots, n_{70}

such that

k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor

and

k

divides

n_{i}

for all

i

such that

1 \leq i \leq 70.

Find the maximum value of

\frac{n_{i}}{k}

for

1\leq i \leq 70.

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