Random Number Theory Problem~USAMO
Posted:
Mon Apr 09, 2007 10:01 pm
by 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]
Posted:
Mon Apr 09, 2007 10:49 pm
by 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.