AIME II Problem 7 2007

[zefuri]

Given a real number [unparseable or potentially dangerous latex formula], let [unparseable or potentially dangerous latex formula] denote the greatest integer less than or equal to [unparseable or potentially dangerous latex formula]. For a certain integer [unparseable or potentially dangerous latex formula], there are exactly [unparseable or potentially dangerous latex formula] positive integers [unparseable or potentially dangerous latex formula] such that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] divides [unparseable or potentially dangerous latex formula] for all [unparseable or potentially dangerous latex formula] such that [unparseable or potentially dangerous latex formula]

Find the maximum value of [unparseable or potentially dangerous latex formula] for [unparseable or potentially dangerous latex formula]

[AuSmith]

I'm assuming it said exactly 70 distinct positive integers. Otherwise, it would have no bearing on the problem.

Find all types of integers [unparseable or potentially dangerous latex formula] with [unparseable or potentially dangerous latex formula].

Since cuberoot is strictly increasing, [unparseable or potentially dangerous latex formula] can take on any integer value from [unparseable or potentially dangerous latex formula] to [unparseable or potentially dangerous latex formula]. Further, it must be divisible by k. Possible [unparseable or potentially dangerous latex formula] are reduced to integers of the form [unparseable or potentially dangerous latex formula] for [unparseable or potentially dangerous latex formula]. Since we have exactly 70 distinct [unparseable or potentially dangerous latex formula], each represented by a [unparseable or potentially dangerous latex formula] value, there must be exactly 70 values for b.

[unparseable or potentially dangerous latex formula].

Therefore, the maximum [unparseable or potentially dangerous latex formula] is [unparseable or potentially dangerous latex formula]

[zefuri]

[unparseable or potentially dangerous latex formula]
Let [unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

That's a range of [unparseable or potentially dangerous latex formula] integral values. We're given [unparseable or potentially dangerous latex formula], so [unparseable or potentially dangerous latex formula].[unparseable or potentially dangerous latex formula].

(edit~mp - fix broken tex)