DropBox now supports screenshots

You must upgrade the client to version 2.4.0 . in order to appear in the settings option screenshots.

Now by pressing printscreen , alt + printscreen , ctrl + alt + printscreen , ctrl + printscreen , a screenshot will be taken and saved to the DropBox folder. Combinations with ctrl allow you to immediately copy the screenshot link to the clipboard.
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On Mac works with Makovsky hotkeys for screenshots.

upd: As the Firz habraiser suggests , this feature is already present in version 2.3.31