Random Number Theory Problem~USAMO

[zefuri]

Let [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] be positive integers such that [unparseable or potentially dangerous latex formula]. Prove [unparseable or potentially dangerous latex formula]

[AuSmith]

Suppose [unparseable or potentially dangerous latex formula] and either [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula].

We are going to use the simple fact that if [unparseable or potentially dangerous latex formula], then [unparseable or potentially dangerous latex formula].

In other words, we formulate the stronger statement that if [unparseable or potentially dangerous latex formula], then [unparseable or potentially dangerous latex formula].

Suppose [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] for some [unparseable or potentially dangerous latex formula].

[unparseable or potentially dangerous latex formula] (N is even)

or equivalently,

[unparseable or potentially dangerous latex formula]

for every even integer N. If [unparseable or potentially dangerous latex formula], we reach a contradiction ([unparseable or potentially dangerous latex formula] cannot be true).

The case where [unparseable or potentially dangerous latex formula] is almost exactly the same.