AIME #1 1994

[stupidityismygam]

sorry brad, the amc 12 problems got boring..

The increasing sequence [unparseable or potentially dangerous latex formula] consists of those positive multiples of [unparseable or potentially dangerous latex formula] that are one less than a perfect square. What is the remainder when the [unparseable or potentially dangerous latex formula] term of the sequence is divided by [unparseable or potentially dangerous latex formula]

[bradp]

[stupidityismygam]

hehe, you lose

that is incorrect

[bradp]

hold on i see where i mixed it up

[Roel]

The sequence looks like this:

[unparseable or potentially dangerous latex formula]

The numbers squared in the even terms of the sequence are obtained by [unparseable or potentially dangerous latex formula] where n is the [unparseable or potentially dangerous latex formula] EVEN term in the sequence.

[unparseable or potentially dangerous latex formula] is the [unparseable or potentially dangerous latex formula] even term in the sequence so: [unparseable or potentially dangerous latex formula]
Now: [unparseable or potentially dangerous latex formula] so the answer is [unparseable or potentially dangerous latex formula]

[stupidityismygam]

[stupidityismygam]

roel wins the grand gold championship :)

now go to the next problem

[zefuri]

this is a number 1...

Let [unparseable or potentially dangerous latex formula] denote the square root of the perfect square 1 greater than each number in the sequence.

We know that [unparseable or potentially dangerous latex formula] if [unparseable or potentially dangerous latex formula] is not divisible by 3 because by Fermat's Little Theorem,

[unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime.

If [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]. We have proved that [unparseable or potentially dangerous latex formula] must not be divisible by [unparseable or potentially dangerous latex formula].

We know that the first term in the sequence is true when [unparseable or potentially dangerous latex formula].

The [unparseable or potentially dangerous latex formula] element of the sequence is found by

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

Thus, we plug in [unparseable or potentially dangerous latex formula] and get [unparseable or potentially dangerous latex formula]. and square it and subtract 1 and divide by 1000 to get [unparseable or potentially dangerous latex formula]