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MATH / Given a real number x, let ⌊x⌋ denote the greatest integer less than or equal to x. For a certain integer k, there are exactly 70 positive integers n₁, n₂,…, n₇₀ such that …

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problem

Given a real number xx, let ⌊x⌋\lfloor x\rfloor denote the greatest integer less than or equal to xx. For a certain integer kk, there are exactly 70 positive integers n1n_1, n2,…,n_2, \ldots, n70n_{70} such that k=⌊n13⌋=⌊n23⌋=⋯=⌊n703⌋k = \lfloor \sqrt[3]{n_1} \rfloor = \lfloor \sqrt[3]{n_2} \rfloor =\cdots= \lfloor \sqrt[3]{n_{70}} \rfloorand kk divides nin_i for all ii such that 1≤i≤701 \leq i \leq 70. Find the maximum value of $\displaystyle\frac{n_i}{k}$ for 1≤i≤701 \leq i \leq 70.
Plain-text mathematical notation (without MathML)
Given a real number x, let ⌊x⌋ denote the greatest integer less than or equal to x. For a certain integer k, there are exactly 70 positive integers n₁, n₂,…, n₇₀ such that
k=⌊root[3](n₁)⌋=⌊root[3](n₂)⌋=⋯=⌊root[3](n₇₀)⌋and k divides n_(i) for all i such that 1≤i≤70.

Find the maximum value of $\displaystyle\frac{n_i}{k}$ for 1≤i≤70.
Original LaTeX notation
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 $\displaystyle\frac{n_i}{k}$ for $1 \leq i \leq 70$.

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
d8f5488f-1b68-56f3-81c0-f68947cf8392
task_key
intermediate~5falgebra--test--d8f5488f-1b68-56f3-81c0-f68947cf8392
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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