Ibanez RG350DX not in tune? Strings not in tune with themselves?

This may sound weird, but I recently bought an Ibanez RG350DX, and the low E string doesn't seem to be in tune with itself. Yes, it sounds awkward, but it seems true. Here's the situation: I tune the low E open string to a piano, right? I get the pitch just right. So I play say, the 5th fret (an A) and I play an A on the piano. The pitch of the open string is perfect, but the A on the string is sharp. And even the octave on the same string doesn't seem to be in tune, it's about a quarter step or so sharp. I've never heard of strings being out of tune with themselves, and this is a very strange, frustrating phenomenon.