USAMO 1999

[zefuri]

Let [unparseable or potentially dangerous latex formula] be a prime and let [unparseable or potentially dangerous latex formula] are integers not divisible by [unparseable or potentially dangerous latex formula], such that
[unparseable or potentially dangerous latex formula] for any integer [unparseable or potentially dangerous latex formula] not divisible by [unparseable or potentially dangerous latex formula].

Prove that at least [unparseable or potentially dangerous latex formula] of the pairwise sums of [unparseable or potentially dangerous latex formula] are divisible by [unparseable or potentially dangerous latex formula].
([unparseable or potentially dangerous latex formula] is the fractional part of [unparseable or potentially dangerous latex formula])

[stupidityismygam]

ok, i was just brainstorming this is what i meant

[unparseable or potentially dangerous latex formula] is an integer so wince p is prime [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula], then [unparseable or potentially dangerous latex formula]

that is all i have...i was hoping someone could own it from there

[makashi]

[stupidityismygam]

since [unparseable or potentially dangerous latex formula], then let [unparseable or potentially dangerous latex formula], so plugging it into the original you get [unparseable or potentially dangerous latex formula] for an integer [unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula] (though i dont think that helps)

we now limit r to be in the range [unparseable or potentially dangerous latex formula] since there will be the same result in (mod p)

so, now basically all we have to worry about with r is [unparseable or potentially dangerous latex formula]

hmm....that is all i have