You said that it duplicated a note. I'm guessing that a problem occurred with that note, corrupting it, and that it is crashing the Notes app each time it tries to open and display that note.
A similar problem can happen with Mail and emails that get corrupted. The solution is the same. Go in by another method and delete the corrupted file.
Is this a tried and true solution to this problem? No. Because I've never heard of this specific problem with Notes. I have heard of Notes spontaneously deleting everything, but not a persistent crash that could not be corrected by a restart or reset; which you seem to have already tried.
If this does not work, the only other idea I have is a clean restore. A rather large pain, so it's best left as a last resort. If the Notes app itself has somehow become corrupt (not impossible) then a restore will be the only way of fixing it.
While I don't think it will help, you could also try Reset All Settings. This has surprised me a few times in the past by fixing things I'd never think it would help. Since all it does is default most of your settings (without affecting account information) it's not as drastic as a restore.
In the end, I tend to recommend people use an alternative app to Notes. At least for important stuff. I like Evernote, but there are tons of apps out there. I use Notes only as an in-between sort of app. A kind of editable clipboard. Anything important goes into other apps with a more reliable history, and more importantly a better backup/restore solution.
While I'm sure you've tried them all, or most, I'll append my Usual Fixes post; in case there is something you (or I) missed.
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The Usual Fixes
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When you have problems with an app it's a good idea to clear it from RAM.
Close/Clear the app:
iOS 6: With the problem app closed, double tap on the Home button to get the multitask bar. Tap and hold on any icon in the bar until they jiggle, then tap the minus sign on the problem app.
iOS 7: Double tap the Home button to get the multitasking display. Swipe up on the problem app's thumbnail image (not the icon).
If that does not work try a restart,
Restart: Hold down the power button until the power off slider appears. Slide it off. When the iPad finishes shutting down, hold down the power button again until the Apple logo appears. Apple recommends you restart the iPad at least once a month.
and then a reset if that does not work.
Reset: Hold down the power and home buttons until the Apple logo appears. The iPad will reboot. This usually takes a bit longer than a restart.
Reset All Settings: Another, somewhat more drastic thing you can do is Reset All Settings. You'll find this under Settings > General > Reset > Reset All Settings. It won't erase any of your content, apps, or accounts; but you will have to go through most of your other settings and re-tweak them back to your liking.
Do not confuse this with Erase All Content and Settings.
More detailed information at:
iOS: Not responding or does not turn on