AIME II Problem 12 2007

[zefuri]

The increasing geometric sequence[unparseable or potentially dangerous latex formula] consists entirely of integral powers of 3. Given that

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

find [unparseable or potentially dangerous latex formula]

[Kurt]

Let [unparseable or potentially dangerous latex formula]

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

Since [unparseable or potentially dangerous latex formula] are powers of 3 themselves, then for some [unparseable or potentially dangerous latex formula], so [unparseable or potentially dangerous latex formula]

Using the second condition, we get:
[unparseable or potentially dangerous latex formula], or, rewriting and using the geometric sum formula:

[unparseable or potentially dangerous latex formula]. After seeing that, with such huge numbers, the 1's are going to barely matter at all:

[unparseable or potentially dangerous latex formula] (approximately).

Rewriting using [unparseable or potentially dangerous latex formula], or:

[unparseable or potentially dangerous latex formula] (again, approximately).

So we have as integer problems:
[unparseable or potentially dangerous latex formula]

Since [unparseable or potentially dangerous latex formula],
[unparseable or potentially dangerous latex formula]
or [unparseable or potentially dangerous latex formula] (as possible integer solutions):

But if [unparseable or potentially dangerous latex formula].

So [unparseable or potentially dangerous latex formula]:

And so [unparseable or potentially dangerous latex formula]

And thus [unparseable or potentially dangerous latex formula]

[zefuri]

wah so dumb forgot about [unparseable or potentially dangerous latex formula] instead of [unparseable or potentially dangerous latex formula]