When I play an A3 on my violin, the A5 string vibrates very loudly, to the point of the A5 string vibrating long after the A3 stops. I can even see the A5 open string vibrate. There is no problem for the E4 and E6 strings, or G3 and D4 strings, or D4 and A4 strings, etc. Should I be worried?
You are describing resonance, which is one of the ways that violinists know a note is in tune. It is usually a good thing, in that the instrument is telling you that the note is in tune (!). Some common resonances in first-position:
1st finger on G-string playing A is in tune => A-string rings 1st finger on D-string playing E is in tune => E-string rings 3rd finger on D-string playing G is in tune => G-string rings 3rd finger on A-string playing D is in tune => D-string rings 3rd finger on E-string playing A is in tune => A-string rings 4th finger on G-string playing D is in tune => D-string rings 4th finger on D-string playing A is in tune => A-string rings 4th finger on A-string playing E is in tune => E-string rings
There are many more such resonances as you play higher up the fingerboard and you can use them to ensure that you stay in tune, even in very high positions.
Composers who want to take advantage of these resonances will often compose in the keys of G,D,A,E, while those looking to minimize this effect might chose choose keys which avoid these notes, such as E-flat, A-flat, C-sharp, or F-sharp.
Paganini famously wrote his first concerto in E-flat major, so the orchestra would not play so loudly and drown him out, and then he would tune his violin so the D-string was set to E-flat, enabling him as the soloist to make use of the extra resonance to great effect.